For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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Proof for trivial sequence shared on facebook

I recently solved the following problem on Facebook that was shared on my timeline. 8 = 56 7 = 42 6 = 30 5 = 20 3 = ? I got the answer 6 by using the following formula, after some inspection, ...
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1answer
20 views

Find when the product would be an integer

The problem: The sequence $\{a_n\}$ is defined recursively by $a_0=1,a_1=\sqrt[19]{2}$ and $a_n=a_{n-1}a_{n-2}^2$ for $n \geq 2$. What is the smallest positive integer $k$ such that the product ...
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20 views

Can we cover the entire plane with the square with area 1/n for each positive integer n?

We have one square with area 1/n for each positive integer n. Is it possible to place these squares in the xy-plane in such a way that they completely cover the entire plane. If Yes, can you describe ...
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14 views

Sum of infinite series of Sech: $\sum_n \text{sech}(x(n-1/2))\text{sech}(x(n+p-1/2))$

I am wondering if anyone recognises the sum $\sum_{n=-\infty}^\infty \frac{1}{\cosh\left(x \; (n+p-\frac{1}{2}) \right) \cosh \left(x\; (n-\frac{1}{2}) \right) }$ ? I am trying to evaluate sums ...
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27 views

Find the interval of convergence $\displaystyle \sum_{n=1}^\infty \dfrac{(3n)!(x^n)}{(n!)^3}$

$\displaystyle \sum_{n=1}^\infty \dfrac{(3n)!(x^n)}{(n!)^3}$ I already found the interval of convergence to be -1/27 < x < 1/27. I am having trouble checking the endpoint of x= 1/27. I need to ...
2
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0answers
36 views

show that, $\left( \sum_{n=-\infty}^{\infty}\frac{(-1)^n}{2n+1}\right)^k=(k-1)\sum_{n=-\infty}^{\infty}\frac{(-1)^{kn}}{(2n+1)^k}$

I saw this sum on a maths paper, it read "Sophomore's dream". I haven't got a clue what is mean, anyway this is the formula. (1) ...
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37 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in ...
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1answer
23 views

Reference for finite sum of $k^{\alpha}$

in this answer, we are given a great formula for $\sum\limits_{k=1}^n k^{\alpha}$ for all real alpha. For a paper I'm writing, I need a reference for a textbook or a paper which contains this result ...
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1answer
34 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - ...
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23 views

How to prove $ \int_{a}^{b}f\left ( x \right )\cot\left ( x \right )dx=2\sum_{n=1}^{\infty }f\left ( x \right )\sin\left ( 2nx \right )dx$

How to prove this equality below $$\int_{a}^{b}f\left ( x \right )\cot\left ( x \right )\mathrm{d}x=2\sum_{n=1}^{\infty }\int_{a}^{b}f\left ( x \right )\sin\left ( 2nx \right )\mathrm{d}x$$ The ...
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1answer
59 views

What is the result of this infinite sum?

I'm trying to find the value of $\mathbf S$ where $$\mathbf S = \sum_{k=1}^\infty \frac{\sin(2\pi k x)}{k}; k \in \mathbb N, x \in \mathbb R^*$$ I had a look to WA which lead me to this result. I ...
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4answers
62 views

Convergence of $\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}$

I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?
3
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2answers
67 views

Fibonacci Pairs

Find all positive integer solutions to $y^2 - xy - x^2 = 1$ and $y^2 - xy - x^2 = -1$ I have written a C++ program to yield some solution for large constants. I must make conjectures based on the ...
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1answer
18 views

Prove that exist a $a_n<0$ by the recursion $a_{n+2}-Pa_{n+1}+Qa_n = 0$ with the condition $\Delta:= P^2-4Q < 0$

Suppose $P, Q > 0$ such that $\Delta:= P^2-4Q < 0$. The sequence $\{a_n\}$ is defined by the recursion $a_{n+2}-Pa_{n+1}+Qa_n = 0$ where both $a_1$ and $a_2$ are real and at least one is ...
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1answer
45 views

Find the sum of the series $((1+p)/(1-p))^k$

$$\sum_{k=1}^\infty \left(\frac{1+p}{1-p}\right)^k$$ where $p \neq 1$. I need to find the sum of this series, could anyone help me?
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2answers
26 views

I do not understand how to solve this

The two sequences of numbers { 1, 4, 16, 64, . . .} and { 3, 12, 48, 192, . . .} are mixed as follows: { 1, 3, 4, 12, 16, 48, 64, 192, . . .}. One of the numbers in the mixed series is 1048576. Then ...
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1answer
40 views

How to prove convergence? $Y_n= X_{n+1}-X_n$

Let $\{X_n\}$ be a sequence of real numbers and let $Y_n= X_{n+1}-X_n$, then $\{Y_n\}$ converges if and only if the sequence $\{X_n\}$ converges.
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2answers
40 views

Is this a Fourier series? What is its sum?

I am trying to find the sum of this series, I cannot identify what kind it is or how to start. Any help is appreciated $$\sum_{n=0}^\infty \left(\frac{\cos(nx)}{n!}\right)$$
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0answers
16 views

Proof Validation: of f(0)=0 given differentiable…

Let $f$ be a differentiable function on an interval $A$ containing $0$, and assume $(x_n)$ is a sequence in $A$ with $(x_n)$ converging to $0$ and $x_n\neq0$ $\forall n\epsilon\mathbb{N}$. Want to ...
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33 views

On a circle, will a point moving in irrational steps ever land at a point it has been at previously?

Suppose you have a circle of radius 1 and a point on that circle. The point now moves in a clockwise direction around the circle in steps with length $\sqrt 2$. Will that point ever land on a point ...
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6answers
260 views

What is the sum of all the Fibonacci numbers from 1 to infinity.

Today I believe I had found the sum of all the Fibonacci numbers are from $1$ to infinity, meaning I had found $F$ for the equation $F = 1+1+2+3+5+8+13+21+\cdots$ I believe the answer is $-3$, ...
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1answer
53 views

Does a convergent sequence in theory ever reach its limit?

Completing a question on the sequence $\{a_n\} = \frac n{2n+1}$. Does $a_n$ in this sequence ever actually get to $\frac12$ officially?
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3answers
70 views

If $\sup \{a_n\mid n\in \mathbb{N}\}=1$ then $\frac{1}{1-a_n}\to\infty$

Suppose $(a_n)$ is a sequence such that $a_n<1$ for all $n$ and $s:=\sup \{a_n\mid n\in \mathbb{N}\}=1$. I want to prove that $\frac{1}{1-a_n}\to\infty$. My initial approach: let $M>0$ and ...
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1answer
51 views

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$?

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$? I have proven that the infinite series converges on $(1,\infty)$. I want to use the Weierstrass M-test to prove this ...
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22 views

A property on finite sequences $1,1,x_2, x_3, \dots, x_{n-1}$ with $x_i \in \{0,1\}$

Consider a finite sequence $$x_0, x_1, \dots, x_{n-1}$$ with $n$ odd, all $x_i \in \{0,1\}$ and in particular $x_0 = x_1 = 1$. Furthermore, assume that the number of nonzero $x_i$ is even and $\leq ...
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3answers
6k views

I have a problem understanding the proof of Rencontres numbers (Derangements)

I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., ...
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3answers
49 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [on hold]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
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33 views

“Indexes the sequence” meaning in the definition of a subsequence

Let $(a_n)$ be a sequence of real numbers, and let $n_1<n_2< n_3 <n_4 <n_5 <···$ be an increasing sequence of natural numbers. Then the sequence $a_{n_1},a_{n_2},a_{n_3},a_{n_4} ,···$ ...
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2answers
64 views

How to evaluate this series using fourier series?

With the help of Hermite's Integral,I got $$\sum_{n=1}^{\infty }\frac{1}{n}\int_{2\pi n}^{\infty }\frac{\sin x}{x}\mathrm{d}x=\pi-\frac{\pi}{2}\ln(2\pi)$$ I'd like to know can we solve this one using ...
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27 views
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1answer
24 views

Limit of a recursive sequence

Given $z_1>0$ and $a >0$ $z_{n+1} = (a+z_{n})^{1/2}$ To show $z_n$ is convergent and find its limit. The general approach would be to use PMI(Induction) to show that there exists an upper ...
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2answers
52 views

What is the sum in 37th bracket…?

I have a series like this $$\left(7^0\right)+\left(7+7^2+7^3\right)+\left(7^4+7^5+...+7^8\right)+\left(7^9+7^{10}+...+7^{15}\right)$$ I want to find the sum in the 37th bracket.Can anyone guys ...
2
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1answer
29 views

Show that, $\sum_{n=1}^{\infty}\left[\frac{\beta(2n)}{n}-\ln\left(\frac{n+1}{n}\right)\right] =…$

valid for all $s\ge 1$ $$\beta(s)=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^s}$$ The particular value of $\Gamma\left(\frac{1}{4}\right)=3.6256099...$ Euler's constant is defined by $$\lim_{n \to ...
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2answers
43 views

Closed form for $\prod_{i=0}^{\infty}(1+x^{2^i})$

I've recently come across the infinite product $\prod_{i=0}^{\infty}(1+x^{2^i})$ and I was wondering if there is a closed form expression for this, or even if it diverges for all non-zero $x$. ...
2
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1answer
56 views

Sequence ratio test.

The ratio test for sequences states that if $a_n$ is non-negative and $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$$ then $\lim a_n = 0$ if $L <1$ and $\lim a_n = \infty$ if $L >1$. The test ...
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1answer
46 views

Integration of a polylogarithm: Is this function known?

I would like to integrate a polylogarithm of a given order $$\int dx \mbox{Li}_{n-1}(x)$$ suppose that the order is $n\le 0$ and $x\in(-\infty,0]$, so the function is bounded. I know that it can be ...
3
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0answers
48 views

I discovered a sequence that should converge to $\pi$ , but how to prove that it really converges to $\pi$? [duplicate]

So yesterday I took paper and pencil and did some constructions of geometrical nature and I discovered a sequence that should converge to $\pi$. It goes like this: Define $a_1=\sqrt{2}$ and for ...
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9 views

Infinite Sum of Incomplete Gamma (with “z” varying)

I am trying to see whether the following series can be "reduced" or re-written in terms of well-known functions. This seems different from other questions on here in that the summation is going ...
0
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1answer
30 views

limit of $\sum_{j=n}^{4n}\binom{4n}j(1/4)^j(3/4)^{4n-j}$ as $n\to\infty$

I want to find out the value of the limit: $$\lim_{n \rightarrow \infty}\sum_{j=n}^{4n} \dbinom{4n}{j} \left(\frac{1}{4}\right)^j \left(\frac{3}{4}\right)^{4n-j}$$ I am not getting any hint: please ...
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4answers
112 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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2answers
91 views

Prove these identities using Jacobi's triple product identity.

I am requesting help with deriving some identities from Jacobi's triple product identity: $$\sum_{n=-\infty}^{\infty}z^nq^{n^2}=\prod_{n\geq 0}(1-q^{2n+2})(1+zq^{2n+1})(1+z^{-1}q^{2n+1})$$ Here is ...
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28 views

Proof of an integration equality and an infinite series equality derived thereof

I need hints on the proof of: $$\int_0^\infty\dfrac{\ln(x)^2}{1+x^2}{\rm{d}x}=\dfrac{\pi^3}{8}$$ and then: $$\sum\limits_{n=0}^\infty\left((-1)^n\dfrac{1}{(2n+1)^3}\right)=\dfrac{\pi^3}{32}$$ ...
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1answer
60 views

Does the following series converges $\sum_{n=2}^{\infty} \frac{1}{n (\ln n)(\ln \ln n)^2} $?

Given series $$\sum_{n=2}^{\infty} \frac{1}{n (\ln n)(\ln \ln n)^2} .$$ To determine whether it is convergent or divergent. I tried with ratio test but it is inconclusive. Cauchy condensation test ...
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56 views

What is the flaw in the proof that the sum of all positive integers equals to $\frac{-1}{12}$? [duplicate]

According to this article: http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html ...the infinite series: $1 + 2 + 3 + 4 + \cdots$ does not actually equal ...
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3answers
43 views

finding the sum of this series $(2n-1)^2(1/2)^n$

$\sum\limits_{n=1}^{\infty}(2n-1)^2(\frac{1}{2})^n$ I know via Wolfram that the sum is 17, but I'm not sure I've ever found the sum of such a series before. Any help is appreciated.
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0answers
17 views

Limit of Recursive sequences [on hold]

Given $Z_1 > 0$ and $a > 0$ $Z_{n+1} = (a+Z_n)^{1/2}$ To show $Z_n$ is convergent and find its limit. The general approach would be to use PMI(Induction) to show that there exists an upper ...
4
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1answer
68 views

Show that, $2\sum_{s=1}^{\infty}\frac{1-\beta(2s+1)}{2s+1}=\ln\left(\frac{\pi}{2}\right)-2+\frac{\pi}{2}$.

The Dirichlet beta function is defined as for Re(s)>0 $$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^s}.$$ Show that, ...
0
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1answer
22 views

How to shift the weighted mean of a monotically increasing series of values

I have a monotonically increasing series of values $X, x\in[0,1] \ \forall i\in1,2,...,60$ the weighted mean of which is defined by $$\bar{x}=\frac{x_1+\sum_{i=2}^{60} i(x_i-x_{i-1})}{x_{60}}.$$ Given ...
4
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2answers
301 views

Series about Euler-Maclaurin formula

The Euler-Maclaurin formula says (from Concrete Mathematics section 9.5) \[ \sum_{a\le{}k< b}f(k)=\int_a^bf(x)dx+\left.\sum_{k=1}^m\frac{B_k}{k!}f^{(k-1)}(x)\right|_a^b+R_m \] where ...
10
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1answer
144 views
+50

if $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$, show that $(\frac{x_{n+1}}{x_{n}}) $ convergent

Let a real positive number sequence $(x_{n})$ such that $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$ and $\sqrt{x_{1}}\geq \sqrt{x_0+1}$. Show that $(\frac{x_{n+1}}{x_{n}}) $ convergent. ...