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

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0
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0answers
15 views

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

It looks to me, by doing numerical simulations, that $$ v_{c} = \lim_{a\rightarrow \infty}\sum_{n=1}^{\infty} a^{-\frac{c}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}} $$ converges to some value bigger than ...
3
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2answers
64 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.
0
votes
0answers
12 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
23 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
46 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
47 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
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2answers
40 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
70 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
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1answer
37 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 ...
4
votes
1answer
38 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 \mathrm{mean\ of} \{a_1,\dots,a_n\} \end{split} ...
0
votes
2answers
34 views

How to find the limit of the sequence

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1X_n$ 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 I ...
5
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1answer
126 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
90 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
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0answers
41 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
52 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
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4answers
41 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
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1answer
36 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
17 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
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2answers
63 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 ...
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1answer
36 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
57 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
34 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
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0answers
27 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
45 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
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3answers
71 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
35 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
34 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 ...
0
votes
1answer
37 views

How to evaluate this infinite sum

I want to find $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx$. From binomial theorem, $(1-x)^{\frac{1}{3}}= \sum_{0}^\infty (-x)^n\binom{\frac{2}{3}}{n}$. Then $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx= ...
1
vote
1answer
18 views

If a,b,c are in AP and $a^2,b^2,c^2$ are in HP, then prove either $a=b=c$ or $a,b,- \frac c2 $ are in GP

As the title says. Although first part of the proof is obvious, I'm still able to prove it. And for the second part, I'm essentially trying to prove $b^2=-c/a$ (which is possible only when c<0 ...
0
votes
1answer
33 views

What is the difference between the limit of a sequence and a limit point of a set?

I always thought they were the same thing. The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point ...
1
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2answers
33 views

I can't understand this difference equation step

I am working on birth-death processes and I can't understand a step that is taken in a proof. The mean of a process is defined as $$\mu(t) = \sum_{n=1}^{\infty}np_n(t)$$ At certain stage in the ...
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2answers
43 views

Probability of ultimate extinction? Need to show that an infinite series is less than $1$

I have the following probability generating function for a branching process - $$G_n(s) = \frac{n}{n+1} + \sum_{r=1}^{\infty}\frac{n^{r-1}}{(n+1)^{r+1}}s^r$$ It says in a book that extinction is ...
1
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3answers
31 views

Can someone help me understand the proof that every cauchy sequence is bounded?

This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I ...
0
votes
2answers
20 views

Serie continuity

If I have $$f(x) = \sum_{n=0}^\infty \frac{1}{1+x^n} \quad x\in\;]1,\infty[$$ How to show that $f(x)$ is continuous ? I think I should use that $$\lim_{n\longrightarrow\infty}f_n(x) = ...
1
vote
2answers
74 views

Don't understand proof that $x_n \rightarrow A$ $\iff$ every subsequence of $\{x_n\}$ converges to $A$

So we are given that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, we have that $|x_n - A| < \epsilon$ and we want to show that for all $\epsilon' > 0$, there ...
0
votes
1answer
30 views

How to find a recursive formula for some sequence

I know how to find a non-recursive formula for a recursively defined sequence. However, now I have this puzzle which gives me a sequence (but not the recursive definition) and challenges me to find ...
0
votes
1answer
26 views

To check whether series is convergent or not

I need to evaluate these two questions $P_1$ : $n$th term is given by $$\frac{(n + 1 )^n}{n^{n+(3/2)}}$$ $P_2$: $n$th term is given by $$n^2x(1-x^2)^n,\quad 0< x < 1$$ For $P_1$ I tried root ...
9
votes
2answers
212 views

To evaluate limit of sequence

How do I evaluate the limit of the following sequence $$a_n = \left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$$ I tried to take log and ...
1
vote
1answer
29 views

What can you say about a series with terms equal to zero after the $n$-th term?

I have a series that converges and that has the terms equal to zero after the $n$-th element. What can you say about the sum of the absolute values? Thanks!
1
vote
0answers
39 views

Evaluating $\sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \ s>0 $

How can we evaluate the following sum. $$ \sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \quad s>0 $$
3
votes
0answers
47 views

How find this sequence recursive relations

Question: Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...
2
votes
1answer
35 views

Sequence $0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi }{2^{n}}\right)$ and let $c_{n}=a_{n}\sin\left(\dfrac{\pi ...
6
votes
0answers
30 views

Product of permutations of consecutive numbers yields arithmetic sequence

Let $n\geq 3$ be an integer, and $a,b$ be positive integers. Let $c_1,\ldots,c_n$ be a permutation of $a,a+1,\ldots,a+(n-1)$, and $d_1,\ldots,d_n$ be a permutation of $b,b+1,\ldots,b+(n-1)$. Is it ...