Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Number of real roots of $3^x+4^x=2^x+5^x$ with proof This equation
$$3^x+4^x=2^x+5^x$$
has two obvious real roots. The question is if it has more real roots than two. A proof is required in any case.
| We consider the equation
$$3^x+4^x=2^x+5^x~~~~(1)$$
Use Lagranges Mean Value Theorem (LMVT) for the function $f(t)=t^x$ for two intervals $(2,3)$ and $(4,5)$. So
$$\frac{3^x-2^x}{3-2}=xt_1^{x-1}, ~~~t_1 \in (2,3)~~~~(2)$$
and
$$\frac{5^x-4^x}{5-4}=xt_2^{x-1}, ~~~t_2 \in (4,5)~~~~(3).$$
By equating (2) and (3), we get ... | {
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"url": "https://math.stackexchange.com/questions/3275626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Eigenvalues with no eigenvectors Can a matrix (example 2x2) with 2 distinct eigenvalues have no eigenvetors?
(since, the A-3I as well as A-1I both are coming out as invertible with -3 and -3 with only 0 vector in the nullspace)?
| By definition, a matrix has eigenvalue $\lambda$ and eigenvector $\mathbf{x}$ when $A \mathbf{x}=\lambda \mathbf{x}$ holds for some $\lambda$ and a non-zero vector $\mathbf{x}$.
To address the matrix example in the comments ...
$$A=\begin{pmatrix} 2 & 1\\ 1 & 2 \end{pmatrix}$$
The eigenvalues are $\lambda_1 =1$ and $\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Asymptotic expansion of maximum of function. Denoted $M(n)$ by maximum of
$$e^{-t}-\left(1-\frac{t}{n}\right)^n \ \ \ t\in [0,n]$$
how to calculate the asymptotic expansion of $M(n)$ as $n\rightarrow \infty$?
| If we consider the function
$$f(t)=e^{-t}-\left(1-\frac{t}{n}\right)^n $$ its first derivative is given by
$$f'(t)=-e^{-t}+\left(1-\frac{t}{n}\right)^{n-1} $$ It cancels at a value
$$t_*=n+(n-1)\, W\left(-\frac{ n}{n-1}e^{-\frac{n}{n-1}}\right)$$ where $W(.)$ is Lambert function.
So $M(n)=f(t_*)$.
Using successive Tay... | {
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"url": "https://math.stackexchange.com/questions/3279719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability that $2$ dice selected from $3$ rolled dice will have sum $7$ If I roll $3$ dice ($6$-sided) and then proceed to choose $2$ of them (discarding the 3rd), what is the probability of the total of those $2$ dice totalling $7$?
| Let's say the dice are blue, green, and red. Each outcome can be represented by the ordered triple $(b, g, r)$. Since there are six choices for each entry, there are $6^3 = 216$ possible outcomes. Assuming the dice are fair, they are equally likely to occur.
Method 1: The favorable outcomes are those in which at le... | {
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"url": "https://math.stackexchange.com/questions/3283971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $3x(2+\sqrt{9x^2 + 3}) + (4x-2)(\sqrt{x^2 - x +1}+1) = 0$ for $x\in \mathbb{R}$
Solve the equation in $ \mathbb{R}$: $$3x(2+\sqrt{9x^2 + 3}) + (4x-2)(\sqrt{x^2 - x +1}+1) = 0$$
I've been tried to solve this question for 3 hours, but can't find out any answers.
Just like I running in the maze, if I represent ... | It's obvious that for $x\leq0$ our equation has no real roots and since
$$\left(4x-2)\sqrt{x^2-x+1}\right)'=\frac{8x^2-8x+5}{\sqrt{x^2-x+1}}>0,$$ we see that $$10x-2+3x\sqrt{9x^2+3}+(4x-2)\sqrt{x^2-x+1}$$ increases for $x>0$ and our equation has one real root maximum.
But $\frac{1}{5}$ is a root and we are done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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distribution of digits in prime numbers I am curious about this: suppose we consider all numbers in base $b$ such that the number of digits $n$ in this range is the same ( eg, in base $10$ it could be $10-to-99$ for $n=2$, or $100-to-999$ for $n=3$, etc; leading digit is non-zero), for the prime numbers in this range, ... | In an odd base, an odd number always has an odd number of odd digits.
Proof
$$\begin{eqnarray}(2n+1)+(2p+1)=2q
\\
(2r+1)+(2s)=(2t+1)
\\
(2u)+(2v)=(2w)\end{eqnarray}$$
By $$(x+y)+z=x+(y+z)\tag{4}$$ and $$a+c+d=a+d+c=c+a+d=c+d+a=d+a+c=d+c+a\tag{5}$$ we have $$\underbrace{(2h+1)+\cdots +(2h+1)}_{\text{2i+1 times}}=2e+1\im... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The integral $\int_{0}^{\infty} \frac{\cot^{-1}\sqrt{1+x^2}}{\sqrt{1+x^2}}dx =\frac{\pi}{2}\ln (1+\sqrt{2})$ At Mathematica the numerical value of the integral
$$\int_{0}^{\infty} \frac{\cot^{-1}\sqrt{1+x^2}}{\sqrt{1+x^2}} dx$$
equals 1.3844.., which is nothing but $\frac{\pi}{2}\ln (1+\sqrt{2})=z$. Also, one of its t... | Let
$$I=\int_{0}^{\infty} \frac{\cot^{-1}\sqrt{1+x^2}}{\sqrt{1+x^2}}dx =\int_{0}^{\infty} \frac{\tan^{-1}(1/\sqrt{(1+x^2})}{\sqrt{1+x^2}} dx$$
Let us use the integral representation of $$\frac{\tan^{-1}(1/\sqrt{1+x^2})}{\sqrt{1+x^2}}=\int_{0}^{1} \frac{dt}{1+x^2+t^2}$$
$$I=\int_{0}^{\infty} \int_{0}^{1} \frac {dt dx}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3286507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $x^{3} = 6+ 3xy - 3 ( \sqrt{2}+2 )^{{1}/{3}} , y^{3} = 9 + 3xy(\sqrt{2}+2)^{{1}/{3}} - 3(\sqrt{2}+2)^{{2}/{3}}$ Solve the system of equations for $x,y \in \mathbb{R}$
$x^{3} = 6+ 3xy - 3\left ( \sqrt{2}+2 \right )^{\frac{1}{3}} $
$ y^{3} = 9 + 3xy(\sqrt{2}+2)^{\frac{1}{3}} - 3(\sqrt{2}+2)^{\frac{2}{3}}$
I j... | import numpy as np
import matplotlib.pyplot as plt
a = pow(2+np.sqrt(2),1/3)
y, x = np.ogrid[-10:10:100j, -10:10:100j]
plt.contour(x.ravel(), y.ravel(), x**3-3*x*y+3*a, [6], colors='r')
plt.contour(x.ravel(), y.ravel(), y**3-3*a*x*y+3*a*a, [9])
plt.grid()
plt.show()
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3286730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
} |
Integrate $\int_a^\infty\frac{\sqrt{x^2-a^2}}{\sinh x}\,dx$ I wonder whether the following integral ($a\in\mathbb{R}$, $a>0$)
$$S(a)=\int_a^\infty\frac{\sqrt{x^2-a^2}}{\sinh x}\,dx$$
admits a closed form (perhaps using some known special functions).
The integral representation of $K_1(z)$ gives just
$$S(a)=2a\sum\limit... | Let's denote:
$$F(a)=\frac{1}{2}+\frac{d}{da} \frac{S(a\pi)}{\pi^2} \tag{1}$$
Remember for later that $S(0)= \frac{\pi^2}{4}$.
From the second expression in the OP we obtain:
$$F(a)=a \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n^2+a^2}} \tag{2}$$
In this answer I found that:
$$\frac{F(a)}{a}=\log 2 +\sum_{k=1}^\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3287065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How many numbers with 6 digits can be formed with the digits 1,2,3,4,5 such that the digit 2 appears every time at least three times? How many numbers with 6 digits can be formed with the digits $1,2,3,4,5$ such that the digit $2$ appears every time at least three times.
My try:
Total numbers: $5^6$
Numbers in which 2 ... | ...or the other way around: $\binom{6}{3}4^3 + \binom{6}{4}4^2 + \binom{6}{5}6^1 +1$. Keep in mind $\binom{6}{4} = \binom{6}{2}$. It's a term longer, but gives you an opportunity to compute sm binomial coefficients.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3287591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Solving 5th degree polynomial Solve the equation, $$6x^5 + 5x^4 − 51x^3 + 51x^2 − 5x − 6 = 0$$(hints: the pattern of the coefficients)
How I attempted this problem is to use the rational root theorem to obtain the factors. The possible roots are $$±(1, 1/2, 1/3, 1/6, 2, 2/3, 3, 3/2, 6)$$
By substituting the values insi... | The "pattern" is that w hen you reverse the order of the coefficients you get the negative of the original polynomial:
$6x^5+5x^4-51x^3+51x^2-5x-6\to-6x^5-5x^4+51x^3-51x^2+5x+6$
When the polynomial has this pattern you can render it this way:
$6x^5+5x^4-51x^3+51x^2-5x-6=6\color{blue}{(x^5-1)}+5\color{blue}{(x^4-x)}-51\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find generating function for the pattern Denote $p(n)$ is the numbers of any size rectangle count on these pattern
Problem :
Find generating function that $p(n)$ is coefficient of function.
I try to read out for some $p(n)$ and get
$p(1)=1$
$p(2)=2\binom{3}{2}-1 = 5$
$p(3)=(2\binom{4}{2}-1) + \binom{3}{2}^{2} - (2\... | We derive a recurrence relation for $p(n), n\geq 1$ and calculate from it the generating function
\begin{align*}
P(x)&=\sum_{j=1}^\infty p(j)x^j\\
&=p(1)x+p(2)x^2+p(3)x^3+p(4)x^4\cdots\\
&=x+5x^2+15x^3+35x^4+\cdots
\end{align*}
In order to go from $p(n-1)$ to $p(n)$ it is convenient to add squares at the diagonal. See ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Splitting a population, probability of 2 people landing in same subpopulation.
$10$ people has been split into $3$ groups $A,B,C$ of $5,3,2$ people respectively. What is the probability that $2$ predetermined people $x,y$ land in the same group?
My attempt:
There is $\binom{10}{5, 3, 2} = \frac{10!}{5!3!2!}$ ways to ... | I also think that you are correct. I get the same answer by just partitioning the remaining eight people ...
$$N(A)= \binom8{3,3,2}+\binom8{5,1,2}+\binom8{5,3,0}
\\= \binom{8}{3}\binom{5}{3} +\binom{8}{1}\binom{7}{5}+\binom{8}{0}\binom{8}{5}
\\=784$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove $x+y+z \ge xy+yz+zx$ Given $x,y,z \ge 0$ and $x+y+z=4-xyz$ Then Prove that
$$x+y+z \ge xy+yz+zx$$
My try:
Letting $x=1-a$, $y=1-b$ and $z=1-c$ we get
$$(1-a)+(1-b)+(1-c)+(1-a)(1-b)(1-c)=4$$
$$-(a+b+c)-(a+b+c)+ab+bc+ca-abc=0$$
$$ab+bc+ca-abc=2(a+b+c)$$
Where $a, b,c \le 1$
is there a clue here?
| Also, we can use $uvw$ here.
Indeed, let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Thus, the condition does not depend on $v^2$ and it's enough to prove our inequality for a maximal value of $v^2$, which happens for equality case of two variables.
Let $y=x$.
Thus, the condition gives $z=\frac{4-2x}{1+x^2},$ where $0\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Solving the following limit without L'Hospital's rule: $\lim_{x\to 0} \frac{\sin(x^2+2)-\sin(x+2)}{x} $
I have been trying to solve the following limit $$\lim_{x\to 0} \frac{\sin(x^2+2)-\sin(x+2)}{x}.$$
I came across the right answer as shown by the steps below, but I would to check if the steps are correct or if so... | You can just sum-to-product it from the beginning and use $\lim_{x\rightarrow 0}\sin(x)/x = 1$:
\begin{align}
\lim_{x\rightarrow 0}\frac{\sin(x^2+2)-\sin(x+2)}{x} &= \lim_{x\rightarrow 0}\frac{2\cos[2+x(x+1)/2]\sin[x(x-1)/2]}{x}
\\ & = \cos(2)\lim_{x\rightarrow 0}\frac{\sin[x(x-1)/2]}{x/2}
\\ &= -\cos(2)\lim_{x\rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Find all extrema of a complicated trigonometric function Problem
Find all local extrema for
$$f(x) = \frac{\sin{3x}}{1+\frac{1}{2}\cos{3x}}$$
Attempted solution
My basic approach is to take the derivative, set the derivative equal to zero and solve for x.
Taking the derivative with the quotient rule and a few cases of ... | It should be $$f'(x) = \frac{(1+\frac{1}{2}\cos{3x})(3\cos{3x})-\frac{3}{2}\sin{3x}\sin {3x}}{(1+\frac{1}{2}\cos{3x})^2}.$$
I like the following way.
Let $x=\frac{2\pi}{9}.$
Thus, we get a value $\frac{2}{\sqrt3}.$
We'll prove that it's a maximal value.
Indeed, we need to prove that
$$\frac{\sin3x}{2+\cos3x}\leq\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate the maximum value of $x^3y + y^3z + z^3x$ where $x + y + z = 4$ and $x, y, z \ge 0$.
Given non-negatives $x, y, z$ such that $x + y + z = 4$. Calculate the maximum value of $$\large x^3y + y^3z + z^3x$$
As an assumption, the maximum value is $27$, occured when $(x, y, z) = (0, 1, 3)$.
I have a guess about a... | Now, use AM-GM:
$$y(x^3+xyz+z^3)\leq y(x+z)^3=27y\left(\frac{x+z}{3}\right)^3\leq27\left(\frac{y+3\cdot\frac{x+z}{3}}{4}\right)^4=27.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3296434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Inequality with Fibonacci numbers $ \sum \limits_{k=1}^{2n+1} (-1)^{k+1} F_k \: \text{arccot} F_k < \frac{\pi+1-\sqrt{5}}{2}$
Prove that $$ \sum \limits_{k=1}^{2n+1} (-1)^{k+1} F_k \: \text{arccot} F_k < \frac{\pi+1-\sqrt{5}}{2}$$ holds for all $n \in \mathbb{N}$.
(The Fibonacci sequence, defined by the recurrence $F_... | Not a full answer.
Let's group even and odd terms in the sum:
$$S_n=\sum_{k=1}^{2n+1} (-1)^{k+1} F_k \operatorname{arccot} F_k= \\ = \sum_{l=0}^{n-1} (F_{2l+1} \operatorname{arccot} F_{2l+1}-F_{2l+2} \operatorname{arccot} F_{2l+2}) +F_{2n+1} \operatorname{arccot} F_{2n+1}$$
It's easy to prove that $F_{2n+1} \operator... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3298554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find all $x$ such that $\sin x = \frac{4}{5}$ and $\cos x = \frac{3}{5}$.
Let
$$
\left\{
\begin{array}{c}
\sin x = \frac{4}{5} \\
\cos x = \frac{3}{5}
\end{array}
\right.
$$
Find all of the possible values for $x$.
My try: By dividing the equations we obtain $\tan x = \frac{4}{3}$ and then $$x = \arctan\frac{4}... | $\cos x=\dfrac35$
$\implies x=2n\pi\pm\arccos\dfrac35$
As for $x>0,\arccos x=\arcsin\sqrt{1-x^2}=\arctan\dfrac{\sqrt{1-x^2}}x$
$\implies x=2n\pi\pm\arcsin\dfrac45$ where $n$ is any integer
Now , $\sin x=\sin\left(2n\pi\pm\arcsin\dfrac45\right)=\pm\dfrac45$
But $\sin x=+\dfrac45$
$$\implies x=2n\pi+\arcsin\dfrac45=2n\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3301407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Alternative methods for solving a system of one linear one non linear simultaneous equations Take the equations $$x+y=5$$ $$x^2 + y^2 =13$$
The most basic method to solve this system is to first express the linear equation in terms of one of the variables and then sub that into the non-linear equation.
But I am curious... | $\DeclareMathOperator{\lcm}{lcm}$
Compute the Gröbner basis of your system. Let us start by writing this with zeroes on the right of the equals signs. \begin{align*}
0 &= x+y-5 \\
0 &= x^2 + y^2 - 13 \text{.}
\end{align*}
We pick a variable ordering. Let us choose $x < y$. (The given system is unchanged by the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3302101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 10,
"answer_id": 3
} |
Given $(x-1)^3+3(x-1)^2-2(x-1)-4=a(x+1)^3+b(x+1)^2+c(x+1)+d$, find$(a,b,c,d)$
Given $(x-1)^3+3(x-1)^2-2(x-1)-4=a(x+1)^3+b(x+1)^2+c(x+1)+d$,
find$(a,b,c,d)$
my attempt:
$$(x+1)=(x-1)\frac{(x+1)}{(x-1)}$$
but this seems useless?
I want to use synthetic division but I don't know how
| It's $$(x+1-2)^3+3(x+1-2)^2-2(x+1-2)-4=(x+1)^3-3(x+1)^2-2(x+1)+4.$$
Can you end it now?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$
What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$.
I have tried:
$$\frac{10!}{(3!\times7!)} \times (-x + 2)^7 \times (x^2)^3 $$
But got an incorrect answer $-15360$.
| By the multinomial formula,
$$(x^2-x+2)^{10}=\sum_{a+b+c=10}\begin{pmatrix}10!\\\!a!\,b!\;c!\!\end{pmatrix}x^{2a}(-1)^bx^b 2^c,$$
and obtaining $x^3$ means $2a+b=3$, whence $c-a=7$, so the possibilities aren't so many:
*
*either $a=0$, so $b=3$, $c=7$;
*or $a=1$, so $b=1$, $c=8$.
Therefore, the coefficient will... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3304628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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How can one integrate $\int\frac{1}{(x+1)^4(x^2+1)} dx$?
How can one integrate $\displaystyle\int\frac{1}{(x+1)^4(x^2+1)}\ dx$?
Attempt:
I tried partial fraction decomposition (PFD) and got lost. The method of u-substitution didn't work for me either.
What else can I do? Can one calculate the integral without PFD?
| Here is a secure and faster method when the fraction has a pole of comparatively high order:
*
*If the pole is not $0$, as is the case here, perform the substitution $u=x+1$ and express the other factors in function of $u$. We have to take care of $x^2+1$. The method of successive divisions yields $x^2+1=u^2-2u+2... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to show the matrix has Rank $\le 5$
I want to show that the following matrix has Rank $\le 5$.
The matrix is
\begin{bmatrix}
2&1&1&1&0&1&1&1\\
1&2&1&1&1&0&1&1\\
1&1&2&1&1&1&0&1\\
1&1&1&2&1&1&1&0\\
0&1&1&1&2&1&1&1\\
1&0&1&1&1&2&1&1\\
1&1&0&1&1&1&2&1\\
1&1&1&0&1&1&1&2
\end{bmatrix}
I found that there is a submatri... | Call column $j$ of this matrix $C_j$, and denote by $\def\2{\mathbf 2}\2$ the column vector with all $8$ entries equal to $2$. Note that for $j=1,2,3,4$ one has $C_j+C_{j+4}=\2$. Thus $C_6=\2-C_2=C_1+C_5-C_2$, and similarly $C_7=C_1+C_5-C_3$ and $C_8=C_1+C_5-C_4$. The last three columns being in the span of the first f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Find $n$ if $\frac{9^{n+1}+4^{n+1}}{9^n+4^n} = 6$ Find $n$ if $$\frac{9^{n+1}+4^{n+1}}{9^n+4^n} = 6$$
In this video they show a shortcut and say $n=-1/2$ without any explanation.
Key observation here is that the geometric mean of $9$ and $4$ is $6$.
It seems numerator and denominator are partial sums of geometric ser... | This is a possible way:
$$\frac{9^{n+1}+4^{n+1}}{9^n+4^n}=6$$
Now: $$3^{2(n+1)}+2^{2(n+1)}=3\cdot 2\cdot 2^{2n}+2\cdot 3\cdot 3^{2n}$$
I obtain: $$3^{2(n+1)}+2^{2(n+1)}=3\cdot2^{2n+1}+2\cdot3^{2n+1}$$
In other words: $$3^{2n+1}(3-2)=2^{2n+1}(3-2)$$
The solution is $n=-\frac{1}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3305369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 4
} |
Find positive integers $a$ such that there are exactly distinct $2014$ positives integers $b$ satisfied $2 \le \dfrac{a}{b} \le 5$.
Find positive integers $a$ such that there are exactly distinct $2014$ positives integers $b$ satisfied $2 \le \dfrac{a}{b} \le 5$.
This is another problem which I couldn't solve in the ... | Your solution isn't quite correct, as I discuss at the end. Instead, as trisct's comment indicates, it's easier to adjust your inequality to bound the $b$ values by multiples of $a$. In particular, with
$$2 \le \frac{a}{b} \le 5 \tag{1}\label{eq1B}$$
First, multiply the left & middle parts by $\frac{b}{2}$ to get
$$b \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Evaluating $\int ^\infty _{0} \frac{x\ln x}{(1+x^2)^2} \,dx$ My first instinct was to evaluate the indefinite form of the integral, which I did by substituting $x=\tan t$, therefore yielding
\begin{align}
\int \frac{x\ln x}{(1+x^2)^2} \,dx
&= \int \frac{\tan t \sec^2 t \ln\tan t}{(1+\tan^2 t)^2} \,dt
&& \text{by substi... | Another way:
Integrate by parts
$$\int\ln x\cdot\dfrac x{(1+x^2)^2}\ dx=\ln x\int\dfrac x{(1+x^2)^2}\ dx-\int\left(\dfrac{d(\ln x)}{dx}\int\dfrac x{(1+x^2)^2}\ dx\right)dx$$
$$=-\dfrac{\ln x}{2(1+x^2)}+\int\dfrac{dx}{2x(1+x^2)}$$
Again $$\int\dfrac{dx}{x(1+x^2)}=\int\dfrac{(x^2+1-x^2)\ dx}{x(1+x^2)}=\int\dfrac{dx}x-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 2
} |
Does $-\frac{1}{2}+\frac{1}{3}-\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\frac{1}{23}+\dots$ converge? Let $d(m)$ be the number of positive divisors of $m$ [including $1$ and $m$]. Let $p_k$ be the $k^\text{th}$ prime number. Consider the series
$$\sum_{k=1}^{\infty }\frac{(-1)^{d(p_{k... | Since $d(n)$ is odd iff $n$ is a square, your sum is equal to the sum of reciprocals of primes, minus twice the sum of reciprocals of primes of the form $n^2+1$. But we have
$$\sum_{p=n^2+1\text{ for some $n$}}\frac{1}{p}<\sum_{n=1}^\infty\frac{1}{n^2+1}<\infty,$$
so your sum diverges to infinity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Finding the limit of the sequence Let $a ∈ R$ Consider $x_{1} = a, x_{2} =
(1+a)/2$
, and by induction $x_{n} :=
(1+x_{n−1})/2$
What is the limit $?$
By replacing $x_{n}$ and $x_{n-1}$ by $l$, we get the limit $l=1$.
So limit should be $1$.
Also the nth term can be represented by $x_{n} = ( a+ 1 + 2 + 2^{2}+......+ ... | As you've already determined,
$$\begin{equation}\begin{aligned}
x_n & = \frac{a + 1 + 2 + 2^2 + \ldots + 2^{n-2}}{2^{n-1}} \\
& = \frac{a - 1}{2^{n-1}} + \frac{2 + 2 + 2^2 + \ldots + 2^{n-2}}{2^{n-1}} \\
& = \frac{a - 1}{2^{n-1}} + \frac{2^{n-1}}{2^{n-1}} \\
& = \frac{a - 1}{2^{n-1}} + 1
\end{aligned}\end{equation}\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\epsilon - N$ proof of $\sqrt{4n^2+n} - 2n \rightarrow \frac{1}{4}$ I have the following proof for $\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$ and was wondering if it was correct. Note that $\sqrt{4n^2+n} - 2n = \frac{n}{\sqrt{4n^2+n} + 2n}$.
$$\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right| ... | You want to show that $\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$. To do this, I would split up the analysis into scratch work and the formal proof.
For the scratch work, you need to find a suitable upper bound. You have done this by showing
\begin{align}
\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3315160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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If $x=\cot6^\circ\cot42^\circ$ and $y=\tan66^\circ\tan78^\circ$, then determine the ratio of $x$ and $y$
If $x=\cot6^\circ\cot42^\circ$ and $y=\tan66^\circ\tan78^\circ$, then
A) $2x=y$
B) $x=2y$
C) $x=y$
D) $2x=3y$
I seriously don’t know where to start. I don’t need the complete answer, but a starting statement which... | as @Ross said
$x = \cot 6 \tan 48 , y = \cot 24 \cot 12$ then we look at
$$\frac{x}{y} = \dfrac{\cot 6 \tan 48 }{\cot 24 \cot 12} = \frac{\cot 6 \tan 24}{\cot 48 \cot 12}$$
$$\cot 6 \tan 24 = \frac{\cos 6 \sin 24}{\sin 6 \cos 24}= \frac{\sin 30 + \sin 18}{\sin 30 - \sin 18} = \frac{1 + 2\sin 18 }{1 - 2\sin 18} $$
I ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I evaluate $\lim _{x\to 0}\left(\frac{x-\sin x}{x\sin x}\right)$ without using L'Hopital or series?
How do I evaluate $\lim _{x\to 0}\left(\frac{x-\sin x}{x\sin x}\right)$ without using L'Hopital or series?
I've tried expanding the variable such as $x = 2y$ or $x = 3y$, but seemed to still get stuck.
| Try the following
$$L=\lim \dfrac{x-x\cos\frac{x}{2}+x\cos\frac{x}{2}-\sin x}{x\sin x}$$ then use the $\sin(x)=2\sin \frac{x}{2}\cos\frac{x}{2}$ now the first two terms limit is zero and the second two terms have a relation with the limit $L$
$$\lim \frac{x \cos\frac{x}{2} -2 \sin\frac{x}{2} \cos\frac{x}{2}}{2x \sin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Help differentating $f(x) = \sqrt\frac{x^2-1}{x^2+1}$ The equation I'm trying to differentiate is, $ f(x) = \sqrt\frac{x^2-1}{x^2+1}$ and I know the answer is meant to be
$$=\frac{\frac{x\sqrt {x^2+1}}{\sqrt {x^2-1}}-\frac{x\sqrt {x^2-1}}{\sqrt {x^2+1}}}{x^2+1}$$
But when I do the working out I get this
$$=\frac{(x^2-... | Going from your second last line to the last line, you put the $x$ factors into the denominator. Instead, you need to leave them in the numerator. This is because the power of $\frac{-1}{2}$ only apply to the expressions in the brackets, but the $x$ factors are outside of the brackets, so they are not affected. Thus, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Getting different answers for an integral: $\frac{1}{2}x-\frac{3}{2}\ln{|x+2|}+C$ vs $\frac{1}{2}x-\frac{3}{2}\ln{|2x+4|}+C$ Problem:
$$\int\frac{1}{2}-\frac{3}{2x+4}dx$$
Using two different methods I am getting two different answers and have trouble finding why.
Method 1:
$$\int\frac{1}{2}-\frac{3}{2x+4}dx$$
$$\int\... | $\ln|2x+4|=\ln|2(x+2)|=\ln|2|+\ln|x+2|=\ln|x+2|+C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}$ Problem:
$$\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}$$
$$\lim_{x \to -\infty} \sqrt{\frac{1}{x^6}}=0$$ so...
$$\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}=\frac{1}{0}$$
The answer is $-1$ and I kn... | Just as $\sqrt{1/x^6}$ goes to $0$, so does $\sqrt{x^6+4}$ go to $\infty$. You cannot substitute just one of these radicals and then simplify, and their unsimplified product is the indeterminate form $0\cdot\infty$ and so cannot be handled directly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3323085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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What is the greatest possible radius of a circle that passes through the points (1, 2) and (4, 5), whose interior is contained Q1?
What is the greatest possible radius of a circle that passes through the points (1, 2) and (4, 5) and whose interior is contained in the first quadrant of the coordinate plane?
I drew ap... | Consider the $2$ points to be $A(1,2)$ and $B(4,5)$. The center of any circle passing through these $2$ points must be perpendicular bisector of $AB$. The slope of $AB$ is $\frac{5-2}{4-1} = 1$, so the slope of the perpendicular bisector is the negative reciprocal, i.e., $-1$. Also, the midpoint of $AB$ is $M(\frac{1+4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3325597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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How can we not use Muirhead's Inequality for proving the following inequality? There was a question in the problem set in my math team training homework:
Show that $∀a, b, c ∈ \mathbb{R}_{≥0}$ s.t. $a + b + c = 1, 7(ab + bc + ca) ≤ 2 + 9abc.$
I used Muirhead's inequality to do the question (you can try out yourself):... | Your proof looks good.
Here's an unsophisticated alternative proof, using only elementary algebra . . .
We don't even need $a,b,c$ to be nonnegative.
As shown below, if $a,b,c\;$are real numbers such that $a+b+c=1$, and if at least one of $a,b,c\;$is between $-1$ and ${\large{\frac{7}{9}}}$ inclusive, then the inequ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3326605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Laurent Expansion of a Composition (with the inverse) Suppose $f(z) = a_1z + a_0 + O(\frac{1}{z})$ as $|z| \rightarrow \infty$. Here $a_1 > 0$. Let $W(z) = z + \frac{1}{z}$.
How do I obtain that the composite function $L(z) = W \circ f^{-1}(z)$ has the expansion $\frac{z}{a_1} - \frac{a_0}{a_1} + O(\frac{1}{z})$ as $|... | Suppose $f(z) = a_1z + a_0 + O(\frac{1}{z})$ as $|z| \rightarrow \infty$ with $a_1>0$. Consider $g(z) = \frac{1}{f(\frac{1}{z})}$. Then $g(0) = 0$ and $g'(0) \neq 0$. So $g$ is analytic near $0$ and has the form $g(z) = g_1z + g_2z^2 + O(z^3)$ near $0$.
$$
\frac{1}{f\left(\frac{1}{z}\right)} = g(z)
$$
$$
\therefore g_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3327509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Evaluate $\int_{0}^{2\pi} \frac{cos \theta}{2 + cos \theta} d\theta$ using the residue theorem My attempt to a solution for $I = \int_{0}^{2\pi} \frac{cos \theta}{2 + cos \theta} d\theta$ is as follows.
On the unit circle we have $z=e^{i\theta} \implies dz = izd\theta \iff d\theta = \frac{dz}{iz}$, and furthermore $co... |
$$I = \int_{|z|=1}^{} \frac{(z+1/z)\frac{1}{2}}{(2+\frac{z+1/z}{2})} \frac{dz}{iz} = \frac{1}{i} \int_{|z|=1}^{} \frac{z^2+1}{z(z^2+4z+1)}dz = \frac{1}{i} \int_{|z|=1}^{} \frac{z^2+1}{z(z+2-\sqrt{3})(z+2+\sqrt{3})}dz = 2\pi \color{red}{i} \ \sum_{j=1}^{2}\text{Res}\left[\frac{z^2+1}{z(z+2-\sqrt{3})(z+2+\sqrt{3})}, z_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3327944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Locus of mid point of $AB$
If the family of lines $tx+3y-6=0.$ where $t$ is variable intersect the lines $x-2y+3=0$ and $x-y+1=0$ at point $A$ and $B.$ Then locus of mid point of $AB$ is
what i try
Intersection of line $tx+3y-6=0$ and $x-2y+3=0$ is
$\displaystyle A\bigg(\frac{3}{3+2t},\frac{6+3t}{3+2t}\bigg)$
and ... | Hint:
$$2k=\dfrac{6+3t}{3+2t}+\dfrac{6+t}{3+t}$$
$$2k-\dfrac32-1=\dfrac{6+3t}{3+2t}-\dfrac32+\dfrac{6+t}{3+t}-1=\dfrac{3}{2(3+2t)}+\dfrac3{3+t}$$
$$2h=\dfrac3{3+t}+\dfrac3{3+2t}$$
Solve the simultaneous equations for $\dfrac1{3+2t},\dfrac1{3+t}$
Finally use $$2(3+t)-(3+2t)=3$$
Optionally we can simplify the result
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3328705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Maclaurin expansion of $\arccos(1-2x^2)$ Maclaurin expansion of $\arccos(1-2x^2)$
This is what I tried.
$f'(x)=2(1-x^2)^{-1/2} \\
f''(x)=2(1-x^2)^{-3/2}+3 \cdot 2 x^2(1-x^2)^{-5/2} \\
f^{(3)}(x)=18x(1-x^2)^{-5/2}+2\cdot 3\cdot 5x^3(1-x^2)^{-7/2} \\
f^{(4)}(x)=18(1-x^2)^{-5/2}+180x^2(1-x^2)^{-7/2}+2\cdot 3\cdot 5\cdot 7... | $arcos(1-2x^2) = \frac{\pi}{2} \sum\limits_{k=0}^{\infty} \frac{(1-2x^2)^{1+2k}(\frac{1}{2})k}{k!+2k\times k!}$
$\forall |1-2x^2|<1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3329518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to solve two inequalities that which simultaneous answers I have two inequalities here, and I must find the answer for both of them simultaneously (joint answer):
$\left\{
\begin{aligned}
\dfrac{2}{x-3} \gt \dfrac{5}{x+6} \\
\dfrac{1}{3} \lt \dfrac{1}{x-2}
\end{aligned}
\right.$
Please give me some hints and I'll... | Let us consider the inequalties
$(1) \quad \dfrac{2}{x-3} \gt \dfrac{5}{x+6} $
and
$(2) \quad \dfrac{1}{3} \lt \dfrac{1}{x-2} $.
Then compute the set $L_1$ of the solutions of $(1)$ and the set $L_2$ of the solutions of $(2)$.
The set of solutions of
$\left\{
\begin{aligned}
\dfrac{2}{x-3} \gt \dfrac{5}{x+6} \\
\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Closed form for the series: $\sum_{n=1}^{\infty}x^{n^2}$ Let $$S=\sum_{n=1}^{\infty}x^{n^2}, \quad|x|<1.$$ It is convergent and the sum is definitely less than $1/(1-x).$
Is there any closed form for $S$ ?
| There is no known closed form of
$$
S(q):=\sum_{n=1}^{\infty}q^{n^2}, \qquad|q|<1.
$$ in terms of elementary functions. A related function, called the Jacobi theta function,
$$
\begin{align}
\vartheta(z; \tau) &= \sum_{n=-\infty}^\infty \exp \left(\pi i n^2 \tau + 2 \pi i n z\right) \\
&= 1 + 2 \sum_{n=1}^\infty \lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Find value for $k$ such that $(x^2-k)$ is a factor for $f(x)=2x^4+(3k-4)x^3+(2k^2-5k-5)x^2+(2k^3-2k^2-3k-6)x+6$ Find value for $k$ such that $(x^2-k)$ is a factor for, $$f(x)=2x^4+(3k-4)x^3+(2k^2-5k-5)x^2+(2k^3-2k^2-3k-6)x+6$$
My Try
Since $x^2-k=0$ when we substitute $x=\pm k$ to $f(x)$ it should be equal to $0.$
But ... | Just do the Euclidean division of the two polynomials. You get
$$f_k(x)=\left(x^2-k\right)\left(2x^2+(3k-4)x+2k^2-3k-5\right)+(2k^3+k^2-7k-6)x+2k^3-3k^2-5k+6$$
And we want the remainder
$$(2k^3+k^2-7k-6)X+(2k^3-3k^2-5k+6)$$
to be the null polynomial. This is possible for the common roots if any of
$$\begin{align}2k^3+k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Four balls are randomly dropped into four boxes Four balls are randomly dropped into four boxes, where any ball is equally likely to fall into
each box.
For a fixed $k = 0, 1, 2, 3$, let $A_k$ denote the event that exactly $k$ boxes are empty. Compute
$P(A_k)$ for each $k = 0, 1, 2, 3$.
I have computed $P(A_0)$ and $P(... | I'll write out the computation "by hand" for $A_2$. There are slicker answers, but this doesn't require any clever observations.
First we drop the first ball, and it doesn't matter where it goes. Then we drop the second ball. There is a $1/4$ chance it lands in the same place as the first, and a $3/4$ chance otherwise.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3335170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Simplifying expression, is it possible? To show that,
$$
(7 + 50^{1/2})^{1/3} + (7 - 50^{1/2})^{1/3} = 2
$$
I am aware of the way where we can "guess" and come up with the following:
7 + 50^(1/2) = (1 + 2^(1/2))^3
7 - 50^(1/2) = (1 - 2^(1/2))^3
Hence simplifying the expression. But can we do it without the guesswork?... | We can use also the following identity.
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc).$$
Let $a=\sqrt[3]{7-5\sqrt2},$ $b=\sqrt[3]{7+5\sqrt2}$ and $c=\sqrt[3]{7-5\sqrt2}+\sqrt[3]{7+5\sqrt2}$.
Thus, since $a+b-c=0,$ we obtain:
$$7-5\sqrt2+7+5\sqrt2-c^3+3\sqrt[3]{(7-5\sqrt2)(7+5\sqrt2)}c=0$$ or
$$c^3+3c-14=0$$ or
$$c^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3337748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Inequality $3\sin^{-1}\left(\frac{\pi^{1/3}}{2^{2/3}}\right)\leq a+b+c$ with some conditions I have a new problem create by myself :
Let $0<a,b,c<\frac{\pi}{2}$ such that $\sin(a)\sin(b)\sin(c)=\frac{\pi}{4}$ then we have :
$$3\sin^{-1}\left(\frac{(\pi)^{\frac{1}{3}}}{2^{\frac{2}{3}}}\right)\leq a+b+c$$
I have trie... | We can use the Tangent Line method here:
$$\sum_{cyc}\left(a-\arcsin\sqrt[3]{\frac{\pi}{4}}\right)=\sum_{cyc}\left(a-\arcsin\sqrt[3]{\frac{\pi}{4}}-\frac{\ln\sin{a}-\ln\sqrt[3]{\frac{\pi}{4}}}{\sqrt{\sqrt[3]{\frac{16}{\pi^2}}-1}}\right)\geq0$$
because $$a-\arcsin\sqrt[3]{\frac{\pi}{4}}-\frac{\ln\sin{a}-\ln\sqrt[3]{\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3338960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Express $\cos 4x$ in terms of only $\sin 2x$ and hence solve the trigonometric equation The question is to express $\cos 4x$ in terms of only $\sin 2x$ and hence solve the trigonometric equation with the restriction of $\theta \in (0^{\circ}, 135^{\circ})$
$$\frac{\cos 5 \theta}{\sin \theta} + \frac{\sin 5 \theta}{\cos... | Rearrange
$$\dfrac{\cos 5 \theta}{\sin \theta} + \dfrac{\sin 5 \theta}{\cos \theta} = 2$$
to get
$$\cos(4\theta) = \sin(2\theta)$$
Then, use $\cos 2x=1-2\sin^2 x$
$$1-2\sin^2(2\theta)= \sin(2\theta)$$
$$\sin(2\theta) = -1, \space \sin(2\theta)=1/2$$
$$\theta = 15^\circ,\space \theta=75^\circ$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3340125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Alligator population Hints only. I feel like I am so close. Population growth: The time rate change of an alligator population $P$ in a swamp is proportional to the square of $P$. The swamp contained a dozen alligators in $1988$ and $2$ dozen in $1998$
$$\frac{dp}{dt} = kp^2$$
$$\int \frac{dp}{p^2} = \int k$$
$$ \frac{... | You have correctly derived that
$$P(t) = -\frac{1}{tk - \frac{1}{12}}\tag{1}$$
and
$$k = \frac{1}{240}\tag{2}$$
so that
$$P(t) = -\frac{1}{\frac{t}{240} - \frac{1}{12}}\tag{3}$$
Your mistake happens when solving for $P(t)=48$. Starting off at
$$48=-\frac{1}{\frac{t}{240}-\frac{1}{12}}\implies\frac{t}{240} - \frac{1}{1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the method to factor $x^3 + 1$? In the solution to a problem, it's stated that
We see that $x^3+1=(x+1)(x^2-x+1)$.
Why is this, and what method can I use for similar problems with different coefficients?
The full problem is
Find the remainder when $x^{81}+x^{48}+2x^{27}+x^6+3$ is divided by $x^3+1$.
| For the full problem, working modulo $x^3+1$ we have $$x^3=-1\implies x^{3n}=(-1)^n\implies x^{81}+x^{48}+2x^{27}+x^6+3=-1+1-2+1+3=2.$$
So no such factorisation is needed. But when it is needed, here's how to do it. By the factor theorem, the fact that $(-1)^3+1=0$ implies $x-(-1)=x+1$ is a factor. So try $x^3+1=(x+1)(... | {
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"source": "stackexchange",
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Finding $\displaystyle \lim_{x \to 2}\frac{\sqrt{x^3+1}-\sqrt{4x+1}}{\sqrt{x^3-2x}-\sqrt{x+2}}$ I came across this question.
Evaluate the limit $$ \lim_{x \to 2}\frac{\sqrt{x^3+1}-\sqrt{4x+1}}{\sqrt{x^3-2x}-\sqrt{x+2}}$$
I tried rationalizing the denominator, substitution, yet nothing seems to cancel out with the d... | $${{\sqrt{x^3+1}-\sqrt{4x+1} \over \sqrt{x^3-2x} - \sqrt{x+2}}
= \left({\sqrt{x^3+1}-\sqrt{4x+1} \over \sqrt{x^3-2x} - \sqrt{x+2}} \right)
\left( {\sqrt{x^3+1}+\sqrt{4x+1} \over \sqrt{x^3+1}+\sqrt{4x+1}} \right)
\left({\sqrt{x^3-2x} + \sqrt{x+2} \over \sqrt{x^3-2x} + \sqrt{x+2}} \right)
=\left({x^3-4x \over x^3-3x-... | {
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"timestamp": "2023-03-29T00:00:00",
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if $4n^2-1=3m^2$ has a positive integer solution, show that $2n-1$ always square Let $n\in \mathbb{N}$, if $4n^2-1=3m^2$ has a positive integer solution, show that $2n-1$ is a perfect square.
For example $n=1$, it is clear.
and $n=13$ because $$ 4\cdot 13^2-1=3\cdot 15^2$$ and $$2n-1=25=5^2$$
| $a^2-1=3b^2$ is a Pell equation, $a^2-3b^2=1$. The solutions correspond to
$$
a_k + b_k \sqrt3 = (2+\sqrt3)^{k}
$$
We have
$$
a_{k+2}=4a_{k+1}-a_k
, \quad
a_0 = 1
, \quad
a_1 = 2
$$
because $2+\sqrt3$ is a root of $x^2 - 4 x + 1$.
Therefore
$$
a_k \bmod 2 = 1,0,1,0,\dots
\\
a_k \bmod 3 = 1,2,1,2,\dots
$$
Write $4n^2-1=... | {
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Cubic diophantine equation with a prime $x^3 + y^3 + z^3 - 3xyz = p$. Question: Find all triple positive integers $(x, y, z)$ so that
$$x^3 + y^3 + z^3 - 3xyz = p,$$
where $p$ is a prime number greater than $3$.
I have tried the following: The equation factors as
$$(x + y + z) (x^2 + y^2 + z^2-xy-yz-zx) = p.$$
Since ... | Strangely enough, the solution is finite.
for the equation:
$$X^3+Y^3+Z^3-3XYZ=q=ab$$
If it is possible to decompose the coefficient as follows: $4b=k^2+3t^2$
Then the solutions are of the form:
$$X=\frac{1}{6}(2a-3t\pm{k})$$
$$Y=\frac{1}{6}(2a+3t\pm{k})$$
$$Z=\frac{1}{3}(a\mp{k})$$
Thought the solution is determined ... | {
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"timestamp": "2023-03-29T00:00:00",
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Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number. There was a problem in a book:
Denote that $y=x+\dfrac{1}{x}$, express $x^7+\dfrac{1}{x^7}$ using $y$.
It's not a hard question, but I find a special sequence:
$x+\dfrac{1}{x}=y\\x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)^3-3\lef... | Krechmar's `A problem book in Algebra' gives an identity if $x+z=p$ and $xz=q$, then
$$x^n+z^n=p^n-\frac{n}{1}p^{n-2} q+\frac{n(n-3)}{1.2}p^{n-4} q^2+...+(-1)^k \frac{n(n-k-1)(n-k-2)....(n-2k+1)}{k!} p^{n-2k} q^k+...$$
Take $z=1/x$ then $q=1$ and letting $x+\frac{1}{x}=y$ we can write
$$f_n(x)=x^n+\frac{1}{x^n}=\sum_{... | {
"language": "en",
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Formulas for Sequences Removing Multiples of 2, 3, and 5 First off, I am a programmer so please excuse if some of the terms I use are not the correct mathematical terms. I was working on devising a function to improve one of my prime number generation algorithms. With this in mind, I first set out to find the formulas ... | This isn’t exactly what you’re looking for, but it’s definitely relevant, and too long for a comment. The formula
$$\bigg\lfloor\frac{n}{2}\bigg\rfloor + \bigg\lfloor\frac{n}{3}\bigg\rfloor + \bigg\lfloor\frac{n}{5}\bigg\rfloor - \bigg\lfloor\frac{n}{6}\bigg\rfloor - \bigg\lfloor\frac{n}{10}\bigg\rfloor - \bigg\lfloor\... | {
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"source": "stackexchange",
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Given $p^2+q^2+r^2+1=s+\sqrt{p+q+r-s}$, solve for $s$
The real numbers $p$, $q$, $r$, $s$ satisfy
$$p^2+q^2+r^2+1=s+\sqrt{p+q+r-s}$$
Find the value of $s$.
I don't even know how to start. Thanks!
| Let $t = \sqrt{p+q+r-s}$, the equation is equivalent to
$$\begin{align} & p^2+q^2+r^2+1 = (p+q+r-t^2) + t\\
\iff & \left(p - \frac12\right)^2+\left(q - \frac12\right)^2+\left(r - \frac12\right)^2 + \left(t - \frac12\right)^2 = 0\\
\implies & p = q = r = t = \frac12\\
\implies & s = p + q + r - t^2 = \frac54\end{align}$... | {
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If $a$, $b$, $c$ are three positive integers such that $a^3+b^3=c^3$ then one of the integer is divisible by $7$ Let on contrary that none of the $a$, $b$, $c$ is divisible by $7$. Then either $a^3\equiv b^3\pmod{7}$ or $b^3\equiv c^3\pmod{7}$ or $c^3\equiv a^3\pmod{7}$.
Now how to go further?
| $7$ is prime. This implies that the multiplicative group $\mathbb{Z}_7^*$ is cyclic and of order $6$. It follows that the image of the action $x\mapsto x^3$ is a subgroup of order $6/3=2$. So there are only two values for $x^3$ mod $7$.
(Plus a third value, $0$, since $0$ was excluded from $\mathbb{Z}_7^*$. But we are ... | {
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Solutions to $\left( \frac{1+3x}{1+2x} \right)^{\frac{1+x}{x}}=2$? I am looking for all non-negative real solutions to
$$
\left( \frac{1+3x}{1+2x} \right)^{\frac{1+x}{x}}=2.
$$
Numerically it seems that there is a unique solution $x \approx 0.4256$. Any ideas how to prove/find it?
| Starting from Andrew Chin's answer and taking logarithms, we need to solve
$$(A-1) \log \left(1+\frac{1}{A}\right)=\log(2)$$
Building the simple $[1,1]$ Padé approximant around $A=4$ gives
$$(A-1) \log \left(1+\frac{1}{A}\right)\sim\frac{3 \log \left(\frac{5}{4}\right)+\frac{ \left(18+800 \log
^2\left(\frac{5}{4}\r... | {
"language": "en",
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"source": "stackexchange",
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how can I find the Side length Two squares inside an equilateral Triangle?
Question: Figure shows an equilateral triangle with side length equal to $1$ . Two squares of side length a and $2a$ placed side by side just fit inside the triangle as shown.
Find the exact value of $a$.
Its an Assessment question from ... | I struggled with this too. But the info is along the bottom. The triangle is equilateral so all angles are $60°$. On the left there is a right-angled triangle - let's call its base $x$.
Triangle 1: Angle = $60°$, opposite = $a$, and adjacent = $x$
On the right there is another right-angled triangle and its base is $1-3... | {
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Equation of circle touching three circles, two of which are intersecting Find the equation of the circle which is tangentially touching three given circles: $x^2+y^2=49$, $x^2+(y-3.5)^2=49/4$, and $y^2+(x-3.5)^2=49/4$.
By tangentially i mean, it touches the smaller two circle externally and the larger one internally.
T... | Say the center of the circle is at $(h,k)$, and it’s radius is $r$.
From the fact that it is touching the “smaller” circles, the distances between its center and their centers are equal to the sum of the radii of the two chosen circles.
This gives us the following equations:
$$\Bigl(h-\frac 72\Bigr)^2 + k^2 = \Bigl(r... | {
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If $a_{k}=2^{2^k}+2^{-2^k}$ then evaluate $\prod_{k=1}^\infty\left(1-\frac{1}{a_{k}}\right)$ If $$a_{k}=2^{2^k}+2^{-2^k}$$ then evaluate $$\prod_{k=1}^\infty\left(1-\frac{1}{a_{k}}\right)$$
I tried using Sophie-Germaine Identity about factorisation for $x^4+4$ but it did not work
| Let $2^{2^{-k}}=b, 2^{2^k}=\dfrac1b $
$1-\dfrac1{a_k}=\dfrac{b^2-b+1}{b^2+1}$
$a_{k+1}=(2^{2^k})^2+(2^{2^{-k}})^2=b^2+\dfrac1{b^2}=\dfrac{b^4+1}{b^2}$
$1-\dfrac1{a_{1+k}}=\dfrac{b^4-b^2+1}{b^4+1}$
Observe that $$(b^2-b+1)(b^2+b+1)=(b^2+1)^2-b^2=b^4+b^4+1$$
and $$(1+b^2)(1-b^2)=1-b^4$$
$$\implies\prod_{m=k}^n\left(1-\df... | {
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"timestamp": "2023-03-29T00:00:00",
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Integration of a rational function involving a quadratic
Evaluate the integral: $$\int\frac{3x}{(1-4x-2x^2)^2}\ dx$$
Here is my work:
*
*Complete the square on the denominator: $$(1-4x-2x^2)=(1-2(2x+x^2+1-1))=(3-2(x+1)^2)$$
*Insert back into denominator. Use substitution $u=x+1$; $x=u-1$; $dx=du$. $$\int\frac{3(u... | Hint:
This formula should be known by heart:
$$\int\!\frac {\mathrm d x}{a^2-x^2}=\frac 1a\,\operatorname{argtanh}\left(\frac xa\right)=\frac 1{2a} \ln\left(\frac{a+x}{a-x}\right).$$
(Similar to $\;\displaystyle\int\!\frac {\mathrm d x}{a^2+x^2}=\frac 1a\,\arctan\left(\frac xa\right) )$
| {
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If $n$ numbers are generated, what is the probability that the product of all those numbers is a multiple of 10? A computer generates random numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ (each has equal probability). If $n$ numbers are generated (with replacement), what is the probability that the product of all those n... | You are assuming that getting $2,4,3,5$ is the exact same as getting $5,2,4,3$ and that somehow the probability of getting $\{2,3,4,5\}$ in any of the 24 orders is the exact same probability of getting $\{3,3,4,5\}$ any of its 12 orders, or getting $\{4,4,4,\}$ in its one order.
You are assuming that getting a set $\{2... | {
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Calculate the coordinates of points on a circle in 3D space Finding the $(x,y)$ coordinates of points along the circumference of a circle in 2D space is fairly easy.
x = r * cos() + Xc
y = r * sin() + Yc
r = radius of circle
(Xc, Yc) = coordinates of circle center
= current angle
I am looking for similar equations... | Think on the intersection between an sphere $x^2+y^2+z^2= r^2$ and a plane $a x + b y + c z = 0$
Solving for $x,y$ we get
$$
x = \frac{-a c z\mp\sqrt{b^2 \left(r^2 \left(a^2+b^2\right)-z^2 \left(a^2+b^2+c^2\right)\right)}}{a^2+b^2}\\
y = \frac{-b^2 c z\pm a \sqrt{b^2 \left(r^2
\left(a^2+b^2\right)-z^2 \left(a^2+b^2+... | {
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Given three positive numbers $a,b,c$. Prove that $\sum\limits_{sym}\frac{a+b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{sym}\frac{a}{bc}})$ .
(A problem due to Mr. Le Khanh Sy). Given three positive numbers $a, b, c$. Prove that
$$\sum\limits_{sym}\frac{a+ b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{s... | After squaring of the both sides we need to prove that
$$\sum_{sym}(a^4b^2-a^4bc+a^3b^3-2a^3b^2c+a^2b^2c^2)\geq0,$$ which is true by Muirhead and Schur.
| {
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Solving $3\sin(2x+45^\circ)=2\cos(x+135^\circ)$ for $x$ between $0^\circ$ and $360^\circ$ Please find the value of $x$ in degree from this equation, with explanation
$$3\sin(2x+45^\circ)=2\cos(x+135^\circ)$$
For $x$ between $0^\circ$ and $360^\circ$.
| Your simplification of
$$3\sin(2x+45^\circ)=2\cos(x+135^\circ)$$
obtaining the left hand side of
$$3\sin2x\cos 45^\circ+3\cos 2x\sin 45^\circ =2\cos x\cos135^\circ-2\sin x\sin135^\circ$$
was very well done.
In fact, $\cos135^\circ=-\cos45^\circ$ and $\sin135^\circ=\sin45^\circ$ and so you can cancel down to
$$3\sin2x+3... | {
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Differentiation using l´Hopital I need to use L´Hopital's rule with this functions:
$$\lim_{x\rightarrow\frac{\pi}{2}} (1-\sin(x))^{\cos(x)}$$
$$\lim_{x\rightarrow\frac{\pi}{4}} (\tan(x))^{\tan(2x)}$$
I take the exponent down using the properties of logarithms and then make the denominator like: $\lim_{x\rightarrow\fra... | Without using L'Hospital:
$$\begin{align}1) \ \lim_{x\rightarrow\frac{\pi}{2}} (1-\sin(x))^{\cos(x)}&=\lim_{x\rightarrow\frac{\pi}{2}} (1-\sin(x))^{\cos(x)}\cdot 1=\\
&=\lim_{x\rightarrow\frac{\pi}{2}} (1-\sin(x))^{\cos(x)}\cdot \lim_{x\rightarrow\frac{\pi}{2}} (1+\sin(x))^{\cos(x)}=\\
&=\lim_{x\rightarrow\frac{\pi}{2}... | {
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"timestamp": "2023-03-29T00:00:00",
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Differentiating $f(x)=\frac{3(1-\sin x)}{2\cos x}$ I'm having trouble to solve the following derivative. I tried to apply the quotient rule, but I cannot get the same result as my textbook.
The function:
$$f(x)=\frac{3(1-\sin x)}{2\cos x}$$
The result I supposed to get:
$$f'(x)=\frac{3}{2}\sec x(\tan x-\sec x)$$
I have... | $$f(x)=\frac{3(1-\sin x)}{2\cos x} =\frac {3}{2} (\frac {1-\sin x}{\cos x})$$
Apply the quotient rule $$ f'(x) =\frac {3}{2}( \frac {-\cos x (\cos x)+\sin x (1-\sin x)}{\cos ^2 x})=$$
$$\frac {3}{2}( \frac {\sin x -1}{\cos ^2 x})=$$
$$\frac {3}{2}( \frac {1}{\cos x} \frac {\sin x -1}{\cos x})=$$
$$\frac {3}{2} \sec x(\... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the thousandth number in the sequence of numbers relatively prime to $105$.
Suppose that all positive integers which are relatively prime to $105$ are arranged into a increasing sequence: $a_1 , a_2 , a_3, . . . .$ Evaluate $a_{1000}$
By inclusion exclusion principle I found the following equation:
$n - \left(... | The number of integers $n \in \{1,2,3,\dots, 105\}$ that are relatively prime to $105$ is
$$\phi(105) = \phi(3 \times 5 \times 7) = 2 \times 4 \times 6 = 48$$
We know that $\gcd(105A + n, 105) = \gcd(n, 105)$.
Since $1000 = 20\times 48 + 40$, we know that there are $20 \times 48 = 960$ numbers relatively prime to $105$... | {
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Given $0 < x \le 1 \le y$. Calculate the minimum value of $\frac{x}{y + 1} + \frac{y}{x + 1} + \frac{x^2 + 3y^2 + 2}{6xy(xy + 1)}$.
Given $0 < x \le 1 \le y$. Calculate the minimum value of $$\large \dfrac{x}{y + 1} + \dfrac{y}{x + 1} + \dfrac{x^2 + 3y^2 + 2}{6xy \cdot (xy + 1)}$$
We have that $$\frac{x^2 + 3y^2 + 2}... | Show that $$\frac{x}{y+1}+\frac{y}{x+1}+\frac{x^2+3y^2+2}{6xy(xy+1)}\geq \frac{3}{2}$$
The equal sign holds if $x=y=1$
| {
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Roots in equation In the equation
$\sqrt{x-2} - \sqrt{6x-11} + \sqrt{x+3} =0$
I got the roots of $x$ being $6$ and $7\sqrt{3}$.
Considering the graph shows only $6$ as being a valid solution, how should I go as figuring this out in the equation itself?
| Let $x-2=p^2,x+3=q^2;p,q\ge0$
As $x+3>x-2, \dfrac pq<1\ \ (1)$
$6x-11=a(x-2)+b(x+3)$
$x=2\implies1=5b$
$x+3=0\implies6(-3)-11=a(-3-2)$
$$\implies p+q=\sqrt{\dfrac{q^2+29p^2}5}$$
$$25(p+q)^2=q^2+29p^2$$
$$2p^2-25pq+12q^2=0$$
$$\dfrac pq=\dfrac{25\pm\sqrt{25^2-4\cdot2\cdot12}}{2\cdot2}=\dfrac{25\pm23}4$$
$\implies \dfrac... | {
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Geometry Rotation and Trigonometry
$A, B, C$ and $D$ are on a line such that $AB=BC=CD$. Also, $P$ is a point on a circle with $BC$ as a diameter. Find $\tan\angle{APB} \cdot \tan\angle{CPD}$.
Let $O$ be the center of $(BPC)$. Let $P$' be the point of intersection of $PO$ and $(BOC)$ again. Then $C$ is the centroid o... |
Let $A\equiv(0,0),B\equiv(1,0),C\equiv(2,0)$ and $D\equiv(3,0)$. Also let $\angle PBC=\theta$
$PB=\cos \theta$. Also, slope of $BP=\tan\theta$.
So, co-ordinates of $P\equiv(1+\cos^2\theta,\sin\theta \cos\theta)$
So, slope of $AP=\frac{\tan\theta}{2+\tan^2\theta}$
$tan\alpha=\frac{\tan\theta-\frac{\tan\theta}{2+\tan^2\... | {
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"source": "stackexchange",
"question_score": "2",
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Markov Chain: Calculating Expectation Reach a Certain Set of States Suppose I have a Markov chain $Z_k$ with $6$ states, as depicted below:
The probability of moving from one node to a neighboring node is $1/2$. For example, the probability of moving from node $1$ to node $2$ is $1/2$ and the probability of moving fr... | For $i\in\{1,\dots,6\}$, let $m_i = E(T_B | Z_t = i)$ be the expected number of steps until reaching $\{3,4,5\}$, starting from state $i$. Trivially, $m_3=m_4=m_5=0$.
To obtain a system of linear constraints, apply first-step analysis (conditioning on the first step out of state $i$). For $i=1$, we have
\begin{align}... | {
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$\sqrt{2} +\sqrt[3]{2}$ is irrational
Prove that $\sqrt{2} +\sqrt[3]{2}$ is irrational.
My attempt: Suppose $\sqrt{2} +\sqrt[3]{2}$ is rational then $\exists$ $x\in \mathbb{Q}$ such that $$\sqrt{2} +\sqrt[3]{2}=x$$ Rewriting the above equation as $$x-\sqrt{2}=\sqrt[3]{2}$$ cubing the above equation gives us $$x^3-3x... | The OP shows great ingenuity with their (now fixed up) answer.
You can also go with Spivak's hint:
From Spivak's Exercise 18.a we know that if $u$ satisfies
$\tag 1 u^n + a_{n-1}u^{n-1} + \dots + a_0 = 0$
for integers $a_{n-1}, \dots, a_0$, then $u$ is either irrational or an integer.
Let
$\quad u = 2^{\frac{2}{6}} +... | {
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"answer_id": 1
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How to find the basis (and cartesian equation(s)) of a sum of two vectorial subspaces? I have the following vectorial subspaces :
U = span $\left(\begin{pmatrix} 2 \\ 0 \\ 1 \\ -2 \end{pmatrix},\begin{pmatrix} 3 \\ 6 \\ 9 \\ -12 \end{pmatrix} \right)$ and V = span $\left(\begin{pmatrix} 0 \\ 2 \\ 1 \\ 0 \end{pmatrix},\... | Since it is a subspace, it pass the origin, so has an equation like:
$$ax+by+cz+dt=0$$
Now put $4$ points in that and solve a system of linear equations to find $a,b,c,d$.
| {
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Change of basis linear transformation Let $S : M_{2,2} → M_{2,2}$ be the linear transformation defined by $S(A) = A + A^T$
.
Consider the bases for $M_{2,2}$ given by:
$$B = \left\{\begin{bmatrix} 1&0\\0&0 \end{bmatrix}, \begin{bmatrix} 0&1\\0&0 \end{bmatrix}, \begin{bmatrix} 0&0\\1&0 \end{bmatrix}, \begin{bmatrix} 0&0... | We have that by matrix representation of the same vector $v$ with respect to the two basis
*
*$v=M_Bv_B$
*$v=M_Cv_C$
with
$$M_B=\left(\begin{array} &1 & 0 & 0 & 0 \\ 0 & 1 &0 &0 \\ 0& 0 &1&0\\ 0 & 0 &0 & 1 \end{array} \right), \quad M_C=\left(\begin{array} &1 & 2 & 0 & -3 \\ 0 & -1 &0 &0 \\ 0& 0 &1&0\\ 0 & 0 &1 & 2... | {
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"url": "https://math.stackexchange.com/questions/3393588",
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"source": "stackexchange",
"question_score": "1",
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Integrate $\sin^4(x)$ Consider the integral
$$\int \sin^4(x)dx$$
Now I could separate the $\sin^4(x)$ into two $\sin^2(x)$ terms and the use power reducing formula
$$\int \sin^2(x)\sin^2(x)dx $$
$$\int \frac{1-\cos(2x)}{2}*\frac{1-\cos(2x)}{2}dx $$
$$\int \frac{(1-\cos(2x))^2}{4} dx$$
$$\int\frac{1}{4}-\frac{2\cos(2x)... | Surely your $8$ should be a $2$ (not to mention each $\cos^2x$ should be $\cos^22x$), giving$$\frac14\int(1-2\cos 2x+\cos^22x)dx=\frac18\int(3-4\cos 2x+\cos 4x)dx=\frac{1}{32}(12x-8\sin 2x+\sin 4x)+C.$$
| {
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Compute $\sum_{n=1}^{\infty} \frac{ H_{n/2}}{(2n+1)^3}$ How to prove that
$$S=\displaystyle \sum_{n=1}^{\infty} \frac{ H_{n/2}}{(2n+1)^3} \quad=\quad \frac{\pi^2G}{4}-\frac{21\zeta(3)\ln(2)}{8}+\frac{\pi^4}{64}+\frac{\Psi^{(3)}(\frac{1}{4})}{512}- \frac{\Psi^{(3)}(\frac{3}{4})}
{512}$$
This problem was proposed by @... | Cornel's way to make it easy. Replace the harmonic number in the numerator by Digamma function, using that $H_{n/2}= \psi(n/2+1)+\gamma$, and then splitting the series using the parity, we have
$$ S=\sum_{n=1}^{\infty} \frac{ \psi(n/2+1)+\gamma}{(2n+1)^3}=\sum_{n=1}^{\infty} \frac{ \psi(n+1)+\gamma}{(4n+1)^3}+\sum_{n=... | {
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"question_score": "8",
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Showing $\lim_{x\to 0}\frac{\sin(x^2\sin\frac{1}{x})}{x}=0$ $$\lim_{x\to 0}\frac{\sin(x^2\sin\frac{1}{x})}{x}=\lim_{x\to 0 }\frac{x^2\sin\frac{1}{x}}{x}=\lim_{x\to 0} x\sin\frac{1}{x}=0$$
Is this solution right?
Thank you very much!
| A simple proof:
$$ \lim_{x\rightarrow0} \frac{\sin(x^2\sin(\frac{1}{x}))}{x} = \lim_{x\rightarrow0} \frac{\sin(x^2\sin(\frac{1}{x}))}{x} \cdot\frac{x}{x}\frac{\sin(\frac{1}{x})}{\sin(\frac{1}{x})} = \lim_{x\rightarrow0} \frac{\sin(x^2\sin(\frac{1}{x}))}{x^2\sin(\frac{1}{x})} \cdot x\sin(\frac{1}{x}) $$
The term $\frac{... | {
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Prove that if $z$ is uni modular then $\frac{1+z}{1 + \bar z}$ is equal to $z$. The expression can be written as
$$\frac{1+z}{\overline {1+z}}$$
Since $z \cdot \overline z=|z|^2$
$$\overline{1+z}= \frac{1}{1+z}$$
As $|z|=1$
So it will become $(1+z)^2$
But the answer is $z$. What am I doing wrong?
| You are asserting that if $z\cdot \overline z=1$ then $(1+\overline z)(1+z)=1,$ too. That is not true. A simple case is $z=1.$
Being unimodular means $z\cdot \overline{z}=1.$ Then $\overline z = z^{-1}.$
So $$\frac{1+z}{1+\overline{z}}=\frac{1+z}{1+z^{-1}}=\frac{1+z}{1+z^{-1}}\cdot \frac z z=\frac{z(1+z)}{z+1}=z$$
Note... | {
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Determining whether three values can be the lengths of a triangle or not
Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers $$ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| $$ can be the lengt... | The hint.
Let $x>y>z.$
Thus, show that
$$\frac{x^2-z^2}{xz}>\frac{x^2-y^2}{xy}$$ and
$$\frac{x^2-z^2}{xz}>\frac{y^2-z^2}{yz},$$ which says that it's enough to check
$$\frac{x^2-y^2}{xy}+\frac{y^2-z^2}{yz}>\frac{x^2-z^2}{xz}$$ or
$$\sum_{cyc}\frac{x^2-y^2}{xy}>0$$ or
$$\sum_{cyc}(x^2z-x^2y)>0$$ or
$$(x-y)(y-z)(z-x)>0,$$... | {
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Number of possible polynomials
Let $a,b,c,d$ be four integers (not necessarily distinct) in the set $\{1,2,3,4,5\}$. Find the number of polynomials of the form $x^4+ ax^3 + bx^2 + cx +d$ which is divisible by $x+1$.
My Try:
Let $f(x) = x^4+ ax^3 + bx^2 + cx +d$, then $f(-1) = 0$. Thus $1+ (b+d) = c+a$. On counting ca... | The number of solutions of $a-b+c-d=1$ for $a,b,c,d\in\{1,2,3,4,5\}$ can be counted as the coefficient of $x$ in the Laurent series
$$ \left(x+x^2+x^3+x^4+x^5\right)^2\left(\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^4}+\frac{1}{x^5}\right)^2,$$
which is also the coefficient of $x$ in
$$ \left(\frac{1}{x^4}+\fr... | {
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quick way to count Suppose we have 10 sticks with length 1-10, respectively. Pick three from them, how many triangles can we form?
I counted one by one and got 50. Is there a quick way? Any help would be appreciated.
| While bruteforcing you can notice a pattern.
$$\begin{array}{c|c}
Odd&Even\\
\hline
4=3+1<3+\{2\}&5=4+1<4+\{2,3\}\\
\hline
6=5+1<5+\{2,3,4\},&7=6+1<6+\{2,3,4,5\},\\ \qquad \quad \ 4+\{3\}&
\qquad \quad \ 5+\{3,4\}\\
\hline
8=7+1<7+\{2,3,4,5,6\},&9=8+1<8+\{2,3,4,5,6,7\}, \\
\qquad \quad \ \ 6+\{3,4,5\},&\qquad \quad \ 7... | {
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Maximum value of $a+b+c$ in an inequality Given that $a$, $b$ and $c$ are real positive numbers, find the maximum possible value of $a+b+c$, if
$$a^2+b^2+c^2+ab+ac+bc\le1.$$
From the AM-GM theorem, I have
$$a^2+b^2+c^2+ab+ac+bc\geq 6\sqrt[6]{a^4b^4c^4} = 6\sqrt[3]{a^2b^2c^2} \\
6\sqrt[3]{a^2b^2c^2} \le1 \\
a^2b^2c^2 \l... | $$4(a+b+c)^2+(a-b)^2+(b-c)^2+(c-a)^2=6(a^2+b^2+c^2+ab+bc+ca)$$
and so $$4(a+b+c)^2+(a-b)^2+(b-c)^2+(c-a)^2\le 6$$
Each of $(a-b)^2,(b-c)^2,(c-a)^2$ is non-negative and so
$$4(a+b+c)^2\le 6.$$
Therefore $a+b+c\le \sqrt \frac{3}{2}$.
| {
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Evaluation of a digamma series involving golden-ratio Let $\varphi =\frac{1}{2} \left(\sqrt{5}+1\right), a=\tan \left(\frac{\sqrt{5} \pi }{2}\right)$, then how can one prove
$$\sum _{n=1}^{\infty } \frac{\psi ^{(0)}(n+\varphi)-\psi ^{(0)}\left(n-\frac{1}{\varphi}\right)}{n^2+n-1}=\frac{\pi ^2 a^2}{\sqrt{5}}+\frac{4 \pi... | We can use the representation
\begin{equation}
\psi(a)-\psi(b)=(a-b)\sum_{p=0}^\infty\frac{1}{(p+a)(p+b)}
\end{equation}
to express the series
\begin{align}
S(s,t)&=\sum _{n=1}^{\infty } \frac{\psi (n+t)-\psi (n+s)}{(n+s) (n+t)}\\
&=(t-s)\sum _{n=1}^{\infty }\sum_{p=0}^\infty\frac{1}{(p+n+t)(p+n+s)} \frac1{(n+s) (n+t)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403711",
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"source": "stackexchange",
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Show that $3^{-n}$ have the interesting property that one half of their repeating binary string is the inverse of the other.
$3^{-n}$ have the interesting property that one half of their repeating binary string is the inverse of the other. Prove it!
$3^{-1}=\overline{0\color{red}{1}}_2$
$3^{-2}=\overline{000\color{r... | Lemma: For any sequence of $k$ digits $A$, where $B$ is the opposite sequence of digits: $$0.\overline{AB}_2 = \frac{A + 1}{2^k + 1}.$$
Proof: Let $x = 0.\overline{AB}$. Then
$$
2^{k} x = A.\overline{BA}.
$$
Therefore,
$$
x + 2^k x = A.\overline{11\ldots 1} = A + 1 \quad \implies \quad x = \frac{A + 1}{2^k + 1}.
$$
So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3407493",
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"source": "stackexchange",
"question_score": "3",
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Number theory problem on divisors!
For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including $1$ and itself). Determine all positive integers $k$ such that $$\frac{d\left(n^2\right)}{d(n)} = k$$ for some $n$.
Please help me solve this number theory problem. I have tried a lot a... | All odd numbers and only them are representable in the form $d(n^2)/d(n)$. But the proof is tricky!
It is clear that no even number is representable as the numerator is always odd.
For $k$ odd we proceed by induction, clearly $1 = d(1^2)/d(1)$ so $1$ is representable. Now suppose that $k>1$ and that all odd integers ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that $\sqrt[n]{n} < (1 + \frac{1}{\sqrt{n}})^2$ for all n in the naturals. I need to prove that $\sqrt[n]{n} < (1 + \frac{1}{\sqrt{n}})^2$ for all n in the naturals.
I started by using Bernoulli's inequality:
$(1+\frac{2}{\sqrt{n}}) < (1 + \frac{1}{\sqrt{n}})^2$
I can say that:
$(1+\frac{2}{\sqrt{n}}) = (1+\frac... | Well for natural (positive) $n$ then $\sqrt[n]{n} \le (1+ \frac 1{\sqrt n})^2 \iff$
$n \le (1 + \frac 1{\sqrt n})^{2n}$
So $(1+\frac 1{\sqrt n})^{2n} \ge 1 + \frac {2n}{\sqrt n}$ and dang... that's not enough.
But lets go one more term.
Remember the reason $(1 + b)^n \ge 1+ nb$ is becase $(1 + b)^n = 1 + nb + C_2b^2 +... | {
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Find all the numbers and aware A $3$-digit number $n$ is said and aware if the last $ 3$ digits of $n ^ 2$ are the same digits of $n$ and in the same order. Find all the numbers and aware
I solved it with some nasty casework:
We must find all integers $0\leq n < 1000$ such that $n^k \equiv n \pmod{1000}$ for any intege... | There is one very nice thing we can use. It is not a coincidence that 0 and 1 were the only numbers that worked for both modulo 8 and 125. We can prove that quite easily, actually. Let $p^k$ be a prime power. Then:
$$
n^2=n\pmod{p^k}
\iff p^k\mid n^2-n = n(n-1)
\iff p^k \mid n \lor p^k \mid n-1
$$
$$
\iff n\in\{0,1\} \... | {
"language": "en",
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Find the minimum of :$P=a+b+c-ab-bc-ca$ soluLet $a,b,c$ be positive real numbers and $a+b+c+abc=4$.
We can rewrite the first equation as $a+b+c=4-abc.$ Then,
\begin{align*}
P&=a+b+c-ab-bc-ca\\&=(4-abc)-ab-bc-ca\\&=4-abc-ab-bc-ca-(a+b+c+1)+(a+b+c+1)\\&=4-(abc+ab+bc+ca+a+b+c+1)+a+b+c+1\\&=4-(a+1)(b+1)(c+1)+a+b+c+1\\&=5+a... | This problem can be taken down with some heavy machinery. From the initial condition, obtain $c=\frac{4-a-b}{ab+1}$. We now want to minimize $$f(a,b)=a+b-ab+\left(\frac{a+b-4}{ab+1}\right)(a+b-1),$$ where $a$, $b$ are on the region bounded by the x and y axes, and the line $x+y\leq 4$. Since $f$ is continuous, it must ... | {
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"url": "https://math.stackexchange.com/questions/3410321",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Calculate the approximate value I am supposed to calculate the approximate value of $$\cos 151^\circ$$ My idea was that I can divide it in the form: $$\cos 90^\circ+ 61^\circ= \cos \frac{\pi}{2}+ \left (\frac{\pi}{3} +\frac{\pi}{180} \right )$$ Then I use the derivation for cosx:
$$-\sin \frac{\pi}{2}\left ( \frac{\pi}... | One way is to use derivatives.
Let $y=f(x) = \cos x$
$dy = -\sin x dx$
Let $x=150^\circ = \frac{2\pi}{3} , dx = 1^\circ\approx 0.0174$
$\cos x = \cos150^\circ \approx -0.8660, \sin x = \sin 150^\circ =0.5$
$y+dy = f(x+dx) = \cos x -\sin x dx = \cos(150^\circ)-\sin(150^\circ)\times0.0174 \approx-0.8747$
$\implies\cos... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Closed form of $\sum^{\infty}_{n=1} \dfrac{1}{n^a{(n+1)}^a}$ where $a$ is a positive integer Recently, I bought a book about arithmetic. I saw a question is like that:
Given that $\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots=\dfrac{\pi^2}{6}$, find the value of $$\dfrac{1}{1^32^3}+\dfrac{1}{2^33^3}+\dfrac{1}{3^... | Let $h(z)= \frac1{(z+1)^k},g(z)=\frac{1}{z^k}$ then
$$\frac{1}{z^k(z+1)^k} -\frac{\sum_{m=0}^{k-1} \frac{h^{(m)}(0)}{m!} z^m}{z^k}- \frac{\sum_{m=0}^{k-1} \frac{g^{(m)}(-1)}{m!} (z+1)^m}{(z+1)^k}$$
is a rational function with no pole thus it is a polynomial and since it vanishes at $\infty$ it is $0$.
$\frac{h^{(m)}(0... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Given the following function how many solutions to $f(x)=0$ are there? I have the following function:
$f: \mathbb{R} \rightarrow \mathbb{R}$
$f(x)=9^x-5^x-4^x$
And I have to find the number of real soltutions (so not necessarily the solutions themselves, just how many are there) for $f(x)=0$ and $f(x)-2 \sqrt{20^x}=0$... | Yes we have that
$$f(x)=9^x-5^x-4^x=0 \iff \left(\frac49\right)^x+\left(\frac59\right)^x=1$$
and since
$$ g(x)=\left(\frac49\right)^x+\left(\frac59\right)^x\implies g'(x)=-\left(\frac49\right)^x\log \left(\frac94\right)-\left(\frac95\right)^x\log \left(\frac95\right)<0$$
$f(x)$ is strictly decreasing and
*
*$\lim_{... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
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Evaluate $\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5$
Evaluate
$$\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5$$
I did this by $$\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5=\lef... | $$\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}=\frac{(1+\sin x+i\cos x)^2}{(1+\sin x)^2+\cos^2 x}$$
$$=\frac{2(1+\sin x)(\sin x+i\cos x)}{2(1+\sin x)}=\cos\left(\frac\pi2-x\right)+i\sin\left(\frac\pi2-x\right).$$
Apply de Movire's Theorem, you can get
$$\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3413692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\sum \frac{x}{x^2+7}\le \frac{3}{8}$ Let $x,y,z>0$ such that $xy+yz+xz=3$. Show that $$\frac{x}{x^2+7}+\frac{y}{y^2+7}+\frac{z}{z^2+7}\le \frac{3}{8}$$
We have: $$x+y+z\ge \sqrt{3\left(xy+yz+xz\right)}=3\rightarrow \frac{3}{8\left(x+y+z\right)}\le \frac{3}{8\cdot 3}$$
Then i will prove $$\sum \frac{x}{3x^... | Also, the Tangent Line method helps.
Let $x=\sqrt3\tan\alpha,$ $y=\sqrt3\tan\beta$ and $z=\sqrt3\tan\gamma,$ where $\{\alpha,\beta,\gamma\}\subset\left(0,\frac{\pi}{2}\right).$
Thus, $$\sum_{cyc}\tan\alpha\tan\beta=1.$$ It follows that $$\gamma=\arctan\left(\frac{1-\tan(\alpha)\tan(\beta)}{\tan\alpha+\tan\beta}\right)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Continued fraction expansion of $\sqrt{d^2+1}$ How can I show that the continued fraction expansion of $\sqrt{d^2+1}$ is $[d, \overline{2d}]$?
And why is the fundamental unit of $\mathbb Q(\sqrt{d^2+1})$ equivalent to $d+\sqrt{d^2+1}$, if $d^2+1$ is squarefree?
| Partial answer/hint. From
$$\sqrt{d^2+1}-d=\frac{1}{\sqrt{d^2+1}+d} \Rightarrow
\sqrt{d^2+1}=d+\frac{1}{d+\sqrt{d^2+1}}\Rightarrow\\
\sqrt{d^2+1}=\color{red}{d}+\frac{1}{\color{red}{2d}+\frac{1}{d+\sqrt{d^2+1}}} \Rightarrow\\
\sqrt{d^2+1}=\color{red}{d}+\frac{1}{\color{red}{2d}+\frac{1}{\color{red}{2d}+\frac{1}{d+\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3427567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find all $n≥1$ natural numbers such that : $n^{2}=1+(n-1)!$ Problem :
Find all $n≥1$ natural numbers such that : $n^{2}=1+(n-1)!$
My try :
$n=1$ we find : $1=1+1$ $×$
$n=2$ we find : $4=1+1$ $×$
$n=3$ we find : $9=1+2$ $×$
$n=4$ we find : $16=1+6$ $×$
$n=5$ we find : $25=1+24$ $√$
Now how I prove $n=5$ only the ... | If $n\ge 6$, dividing $n^2-1=(n-1)!$ by $n-1$ we get
$$n+1=(n-2)!\ge(n-2)(n-3)(n-4)=n^3-9n^2+26n-24$$
Define
$$\begin{align}f(n)&=n^3-9n^2+25n-25\\
&=(n-5)(n^2-4n+5)\\
&=(n-5)((n-2)^2+1)\end{align}$$
and $f(n)>0$ for $n\ge 6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3427795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Find the units digit of $572^{42}$ The idea of this exercise is that you use the modulus to get the right answer.
What I did was:
$$572\equiv 2\pmod {10} \\
572^2 \equiv 2^2 \equiv 4\pmod{10} \\
572^3 \equiv 2^3 \equiv 8\pmod{10} \\
572^4 \equiv 2^4 \equiv 6\pmod{10} \\
572^5 \equiv 2^5 \equiv 2\pmod{10} \\
572^6 \... | Consider $2^k$ modulo $10$.
You will notice that it enters a repeating pattern: $2,4,8,6,2,4,8,6,2,\dots$
Where $2^k\pmod {10} \equiv \begin{cases}2&\text{when}~k\equiv 1\pmod 4\\ 4&\text{when}~k\equiv 2\pmod 4\\8&\text{when}~k\equiv 3\pmod 4\\6&\text{when}~k\equiv 0\pmod 4\end{cases}$
Now, consider the exponent in thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3428727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
If a variable chord of the hyperbola subtend a right angle at the centre, find the radius of the circle it is tangent to
If a variable line $x\cos\alpha+y\sin\alpha=p$ which is a chord of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, $(b>a)$ subtend a right angle at the centre of hyperbola,then prove that it al... | Let $A=(x_1,y_1)$ and $B=(x_2,y_2)$ be the intersections of line $\cos+\sin=$ with the hyperbola. We have $\angle AOB=90°$ if $x_1x_2+y_1y_2=0$. Substituting there
$$y={p-x\cos\alpha\over\sin\alpha}$$
we thus obtain:
$$
\tag{*}
x_1x_2-(x_1+x_2)p\cos\alpha+p^2=0.
$$
But $x_1$ and $x_2$ are the solutions of the equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3429601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the number of natural solutions for $x_1 +x_2 + \cdots + x_k = n$, with $ x_i \notin 3\mathbb{N}$. Find the number of natural solutions for $$x_1 +x_2 + \cdots + x_k = n,$$ with the constraints $x_i \notin 3\mathbb{N}$ for $i=1,2,\dots,k$.
My Attempt:
the generating function of the equation is:
$f(x) =(x+x^2)(1+... | The generating function is
$$
\left(\frac{x+x^2}{1-x^3}\right)^k\tag1
$$
since each variable can take values in the exponents of
$$
\frac{x+x^2}{1-x^3}=x+x^2+x^4+x^5+x^7+x^8+\dots\tag2
$$
Compute the coefficient of $x^n$ in $(1)$:
$$
\begin{align}
\left[x^n\right]\left(\frac{x+x^2}{1-x^3}\right)^k
&=\left[x^{n-k}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.