For questions about recurrence relations, convergence tests, and identifying sequences

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5
votes
2answers
43 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
1
vote
1answer
36 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
4
votes
3answers
80 views

How to compute $\sum_{n=1}^{\infty}\arctan{\frac{3n^2}{2n^4-1}}$

I find this problem on facebook group. Is it possible to find exact value of $\sum_{n=1}^{\infty}\arctan{\frac{3n^2}{2n^4-1}}$. I think this is not telescope sum. And wolfram alpha can not find it. ...
1
vote
2answers
38 views

Absolutely convergent but not convergent

Here, Lemma $2.1$ states that A normed space $X$ is complete if and only if every absolutely convergent series is convergent. I would like to know a series which is absolutely convergent but ...
0
votes
0answers
37 views

How to calculate the series in the modified form?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^nx^n}{1-x^n} $$ Link: how to calculate the series
0
votes
1answer
32 views

how $\prod\limits_{i=1}^{n} (2k-1)/2= (2n)!/{(4^n)n!} = (2n-1)!/[{2^{2n-1}}(n-1)!]$? [on hold]

We know $Γ(n+1/2)=(n-1/2)!= Π (n-1/2)= √π \cdot \prod\limits_{i=1}^{n} (2k-1)/2$ hence $Γ(n+1/2) = (2n)!/{(4^n)n!}√π = (2n-1)!/[2^{2n-1}(n-1)!]\sqrtπ$ But I need the answer of above question to prove ...
3
votes
2answers
40 views

Finding meeting point of discreet functions [on hold]

[rewording per required] I have a sum of arithmetic series $n(1+n)/2$, and a linear function $nk$. If I want to find the meeting point of their sum with $2^m$. Is there a way better than brutal ...
2
votes
2answers
43 views

If $\{x_n\}$ is a sequence in $\mathbb{N}$ and $x_n \rightarrow x$, prove there exists $N$ such that $x_n = x$ for $n \geq N$

Since $x_n \rightarrow x$, we know that for all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - x| < \epsilon$. We want to show that for some $\epsilon ...
1
vote
2answers
40 views

Interchanging the order of a double infinite sum

I'm stuck at a proof of Wald's first equation about interchanging the order of a double infinite sum: Suppose $X_n \ge 0$ and $1_{\{\cdot \}}$ be indicator function. $$ \sum_{n=1}^\infty ...
1
vote
1answer
26 views

Hypergeometric Series Convergence

For the hypergeometric series $\sum_1^\infty $ $(a)_n (b)_n \over(c)_n n!$, I am looking for help proving that the series converges for $a+b-c<0$. I can understand divergence for different ...
2
votes
2answers
39 views

Substitution on Series

The technique of substitution is a prevalent one in mathematics. It can be used in so many branches spanning from algebra to calculus. I appear to run into a problem, however, when trying to use ...
0
votes
1answer
15 views

Sum of diminishing series with constant addition

I am not sure how to derive the formula for this example, although I suspect something from the annuity pension (not sure if that is the correct English word) formulas. I have a start value of 1000. ...
0
votes
2answers
39 views

How to PROVE there are only finite number of sub limit in this sequence that not converge

for example, let $A,B,C\:\in \mathbb{R}\:$ be some constants, and $$ a_n=\begin{cases} A, & n=3k-2,\ k\in \mathbb{N} \\ B, & n=3k-1,\ k\in \mathbb{N} \\ C, & n=3k,\ k\in \mathbb{N} ...
1
vote
1answer
48 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
3
votes
1answer
54 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
7
votes
2answers
117 views

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

Can the residue theorem prove this? $$\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$$
0
votes
0answers
33 views

Convergence of $\sum_{n=1}^{\infty} a^{-\frac{1}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}}$ as $a\to\infty$

It looks to me, by doing numerical simulations, that $$ v_{c} = \lim_{a\rightarrow \infty}\sum_{n=1}^{\infty} a^{-\frac{1}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}} $$ converges to some value bigger than ...
4
votes
2answers
111 views

How to prove $\sum_{n=1}^{\infty}\frac{1}{16n^2-16n+3}=\frac{\pi}{8}$

Wolfram alpha computes $$\sum_{n=1}^{\infty}\frac{1}{16n^2-16n+3}=\frac{\pi}{8}$$ But I don't have any idea to prove this. Thank you.
-1
votes
1answer
18 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
1
vote
1answer
30 views

convergence of the series to inf or not

let $a_n = \dfrac{e^{-(1/2) \times a^2 \times\log(n) }}{a\sqrt{2\pi \log(n)}} $, $a$ is a constant, and the question is if $S_n = \sum a_n$ converge to a finite number. I wonder if I should ...
0
votes
2answers
48 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
0
votes
2answers
48 views

Question about convergence of sequences

Given a sequence $x_n$ = $\sqrt{1}$ , $-\sqrt{1}$,$\sqrt{2}$,$-\sqrt{2}$... If $$y_n = \frac{x_1 + x_2 + x_3 + x_4 + \ldots +x_n}{n} $$ Then sequence $y_n$ is 1.Monotonic 2.NOT bounded ...
0
votes
2answers
43 views

Proving only the summation part of Cauchy-Schwarz

Can you prove only the summation part of Cauchy-Schwarz? What I mean is that $$\Bigl(\sum a_i b_i\Bigr)^2 \leq \sum a_i^2 \sum b_i^2.$$ I only want to show it for the case where $a_i , b_i \geq 0$ ...
2
votes
2answers
84 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
0
votes
1answer
41 views

Square Root of the Product of Convergent Series Converges

How do I go about proving that Square Root of the Product of Convergent Series Converges, where both are greater than 0. So $\sum \sqrt{x_n y_n}$, where $\sum x_n$ and $\sum y_n$ converge, and each ...
8
votes
1answer
55 views

Recursively appending mean to list: Is there a closed form?

I'm pondering the following sequence: $$\begin{equation} \begin{split} a_1 & = b \\ a_{n+1} & = c\frac{1}{n}\sum_{k=1}^{n}a_k = c \times \text{mean of } \{a_1,\dots,a_n\} \end{split} ...
0
votes
2answers
43 views

How to find the limit of the sequence given by $X_{n+1} = \frac12 X_n + \frac1{X_n}$

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1{X_n}$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still ...
5
votes
1answer
130 views

Closed form formula for $2^{2^1}+2^{2^2}+…+2^{2^n}$

So, the series's sum is of below form: $2^{2^1}+2^{2^2}+...+2^{2^n}$ This series is an intermediate work of an bigger problem {So I am concerned with whether ...
2
votes
5answers
91 views

How to determine if the sequence $a_n= (30+12\arctan(n!))/6^n$ is divergent or convergent

$$a_n= \frac{30+12\arctan(n!)}{6^n}$$ Not sure where to start, I know at infinity arctangent tends towards $\frac{\pi}{2}$. I also know I'm supposed to find the limit but not sure how to start, the ...
-2
votes
0answers
48 views

How to prove that the series converges?

Let us suppose that $\vert a_n\vert$ is a real sequence, and set $S_n=\sum_{k=1}^{n}a_k$ and $\sigma_n=\frac{1}{n+1}\sum_{k=1}^{n}S_k$. How can we show that if the series $\sum_{n=1}^{\infty}\vert ...
3
votes
2answers
54 views

Prove two identities relating to series

Show that: $$ (1). \sum_{n=1}^{\infty}\ln\big(\cos \frac{x}{2^n}\big)=\ln \frac{\sin x}{x} $$ $$ (2). \sum_{n=1}^{\infty}\frac{1}{2^n}\tan \frac{x}{2^n}=\frac{1}{x}-\cot x $$ Thank you in advance. ...
0
votes
0answers
7 views

minimal polynomial of a LFSR sequence

I encircled the problem in the figure below. My question is, why $m(x)$ must have degree of at least $u$, why not it has a degree less than $u$? Maybe this is trivial, but I cannot wrap my mind ...
0
votes
4answers
42 views

Infinite Sum of 1/Polynomial

I'm trying to solve this equation: $$\sum_{k = 0}^{\infty}\dfrac{1}{(k+1)(k+3)}$$ Original image at http://i.imgur.com/wXZFxn0.png I attempted to find the sums of $\sum_0^∞\frac{1}{k+1}$ and ...
2
votes
1answer
39 views

Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
1
vote
2answers
19 views

Blocks of Pyramid Pattern Expression [duplicate]

There is a pattern following, and trying to find the algebraic expression Each layer (from the top). Diagram. So the first layer has 1, second has 4, third has 9, and the fourth has 16. That's ...
1
vote
2answers
64 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
-3
votes
1answer
39 views

Can someone help me prove that this sequence converges? [on hold]

I'm having difficulty trying to prove this, mostly because I don't understand the process. There was a proof similar to this in my textbook where they proved if Sn converges to s then lim 1/Sn=1/s and ...
0
votes
1answer
18 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
2
votes
2answers
55 views

let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then

let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then (a). The sequence $(a_n)$ may be unbounded. (b). The sequence ...
0
votes
1answer
11 views

Sequence Convergence when using a forgetting factor $\lambda$

I'd like to know how I can find the convergence formula of the following sequence $x_{i} = \lambda x_{i-1} + y$ with $\lambda \in (0,1)$, $y$ a positive scalar and initially starting from $0$, so ...
7
votes
1answer
63 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow ...
-1
votes
1answer
36 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
0
votes
0answers
29 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
2
votes
2answers
47 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
0
votes
1answer
15 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
1
vote
3answers
87 views

Value of $\sum_{n=0}^{\infty} \exp(-bn^a)$

What is the value of $$ S = \sum_{n=0}^{\infty} \exp(-bn^a) $$ where $a>0$ and $b>0$? I know that $$ \int_0^{\infty} \exp(-bx^a) \, \mathrm{d}x=\frac{1}{ab^{1/a}}\Gamma(1/a) $$ but cannot ...
2
votes
1answer
36 views

Prove that if $\{a_n\}$ is a sequence of natural numbers such that $\{ a_n^{-1}\}$ is an arithmetic sequence then all $a_i$ are equal

Prove that if $\{a_n\}$ is a sequence of natural numbers such that $\left\{\frac{1}{a_n}\right\}$ is an arithmetic sequence then all the $a_n$s are equal. I have no clue where to begin from, ...
0
votes
4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
0
votes
1answer
27 views

Series convergence issues

I want to show that $$\sum_{k=0}^\infty \frac{1}{1+x^n}$$ is $C^1$ on ($1,\infty)$ To do so, I want to show uniform convergence of $\sum_{k=0}^\infty \frac{d}{dx}\frac{1}{1+x^n} = \sum_{k=0}^\infty ...
0
votes
1answer
37 views

Convergence of a sequence involving integral

Consider $f:[-\pi, \pi] \to \mathbb{C}$ is analytic (infinite differentiable) and periodic. Define $a_n:= \frac{1}{2 \pi}\int_{-\pi}^\pi f(x) e^{-inx}dx$ (the Fourier coefficient of $f$). Show ...