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

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1answer
23 views

Probability of drawing an element from a countably infinite sequence

Consider a sequence containing $A$ and $B$ where, starting at $n=0$, there are $2^n A$'s followed by $2^{n+1} B \ $'s, so the sequence begins $$A, B, B, A, A, B, B, B, B, A, A, A, A, B, B, B, B, B, ...
0
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0answers
10 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
2
votes
2answers
15 views

Show that $\lim_{n\to\infty}\sum_{k=1}^n\bigl|k\bigl(f\bigl(\frac{1}{k}\bigr)-f\bigl(-\frac{1}{k}\bigr)\bigr)-2f'(0)\bigr|$ exists

Suppose $f\in C^3[-1,1]$, show that $$\lim_{n\to\infty}\sum_{k=1}^n\left|k\left(f\left(\frac{1}{k}\right)-f\left(-\frac{1}{k}\right)\right)-2f'(0)\right|$$ exists. I realized that ...
1
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1answer
10 views

Show that the series $∑_{m=1}^{∞}((r^{-m})/(2^{m}-1))$ is convergente for some some positive integer $r>0$

The Erdős-Borwein Constant can be found in http://mathworld.wolfram.com/Erdos-BorweinConstant.html My question is : Show that the series $$∑_{m=1}^{∞}((r^{-m})/(2^{m}-1))$$ is convergente for some ...
2
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0answers
30 views

An infinite series, in which each term is a definite integral of a complicated transcendental function

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the infinite series is an integral whose integrand is a complicated transcendental function involving ...
6
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1answer
85 views

Find the values of the positive integers $n$ such that: $\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$ is positive integer

My question is as follow: Find the values of the positive integers $n$ such that: $$\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$$ is positive integer. I can see that for $n=1$ (among some ...
2
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1answer
36 views

Why can't the definition of convergence be alterted to this one?

I am trying to find out of a seqence with the following property is convergent: Let $(r_n)$ be a sequence of real numbers. Suppose there is a number $r\in\mathbb{R}$ such that for any ...
0
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1answer
35 views

sequence defined by norm

Let $(u_n)_n$ be defined by: $\quad \begin{cases}u_1=1 & \\ \\ u_n=\left( \sum\limits_{k=1}^{n}u_{k}\right)^{\frac{1}{2}} & \end{cases} $ Show that $u_{n}\to +\infty$ and $u_n \simeq ...
2
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1answer
34 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{2b!}$, Then prove that the value of ...
-1
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1answer
75 views

What's the limit of this sum $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ [on hold]

Let $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ and s be a complex variable . $s=\sigma +it $ where $\sigma ,t \in\mathbb{R} $ , Note :I edit the ...
0
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1answer
26 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
1
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1answer
45 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
1
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1answer
41 views

Elementary proof $\sum_{n> N}\frac{1}{n^2} < 1/N $

Is there an elementary proof (not using integrals) $$\sum_{n> N}\frac{1}{n^2} < 1/N $$ for this sum?
4
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0answers
42 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
1
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4answers
96 views

How to prove $ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$?

How to prove that $\displaystyle \lim_{n\longrightarrow\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$? I suppose some bounds are nedded, but the ones I have found are not sharp enough (changing ...
2
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2answers
58 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
2
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1answer
43 views

On a $\epsilon$-$n$ proof of a limit of a sequence of functions.

Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t< (k+1) \delta_n$. Define $$ \psi_n(t) = ...
3
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1answer
84 views

Adding Two Power Series if their bounds are different

I have the following product $$\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n\sum_{n=0}^{\infty}\frac{a_nx^{n+2}}{n!}$$ Where $a_n$ is an arbitrary coefficient. If I factor out $x^2$ ...
4
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2answers
71 views

Finding $\lim_{x\rightarrow 0.5^{-}}\sum_{n=0}^{\infty }(-1)^n\left(\frac{x}{1-x}\right)^n$

Finding $$\lim_{x\rightarrow 0.5^{-}}\sum_{n=0}^{\infty }(-1)^n\left(\frac{x}{1-x}\right)^n$$ When I want to use the geometric series, I had a problem with $(-1)^n$ so I stoped.
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0answers
33 views

Clarifying a step in an integration solution

In the accepted answer here, the first two steps in computing the integral are \begin{align} \mathcal{I} =&\frac{1}{2}\int^\infty_0\ln(1-e^{-2x})\ln\left(\frac{x^2}{\pi^2+x^2}\right)\ {\rm d}x\\ ...
2
votes
1answer
41 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
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0answers
25 views

Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
2
votes
1answer
50 views

sum of infinte series with exponential and factorial terms

Want to sum the following series: $$ \sum_{t=1}^\infty e^{-tk} \frac{(tk)^t}{t!} $$ where $k$ is an integer $>0$.
1
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1answer
35 views

Series with $n$th term having integer raised to the power of $n$ in the denominator

$$ 1+\frac{4}{6}+\frac{4\cdot5}{6\cdot9}+\frac{4\cdot5\cdot6}{6\cdot9\cdot12}+\cdots $$ I could reduce it to $n$th term being $\dfrac{(n+1)\cdot(n+2)}{n!\cdot3^n}$. Took me an hour just to get ...
5
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2answers
55 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
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1answer
41 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 ...
6
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4answers
145 views

How to compute $\sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)$

I find this problem on facebook group. $$\mbox{Is it possible to find exact value of}\quad \sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)\ {\large ?}. $$ I think this is ...
1
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1answer
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
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0answers
44 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
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1answer
34 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
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2answers
49 views

Finding meeting point of discreet functions [on hold]

[added context per advised] Hi, I am self-studying number theory, using text of aata, Beezer, 14 and elementary number theory, Pan, 91. So far I have finished group theory but not yet ring and field. ...
2
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2answers
47 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 ...
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2answers
42 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
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1answer
27 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
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2answers
43 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
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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
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2answers
42 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
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2answers
56 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
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1answer
56 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
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2answers
125 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
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0answers
41 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
115 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.
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1answer
19 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
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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
50 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
47 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
92 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
42 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
58 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} ...