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

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14
votes
3answers
486 views

Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$

Compute the limit $$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
2
votes
1answer
48 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
2
votes
3answers
518 views

how to find a cluster point of $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$

i am tryint to find a cluster point of this sequence, but i am having difficulties in definitions. the sequence is this: $(a_{n})_{n \in \Bbb{N}}$ with $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$ the ...
2
votes
5answers
403 views

Infinite sum and integral

I have a question regarding a sum. Is the following expression finite and can be calculated? $$\lim_{a\to\infty}\frac{1}{a}\sum_{b=0}^a \left(\frac{b}{a}\right)^2$$ Could I also approximate the sum ...
3
votes
2answers
187 views

Prove that $\sum \frac{a_n}{a_n+3}$ diverges

Suppose $a_n>0$ for each $n\in \mathbb{N}$ and $\sum_{n=0}^{\infty} a_n $ diverges. How would one go about showing that $\sum_{n=0}^{\infty} \frac{a_n}{a_n+3}$ diverges?
0
votes
1answer
46 views

Find out the length of a recurrence

I have this rules for creating a list of numbers: $x/2$ if $x$ is even, repeat $3x+1$ if $x$ is odd, repeat if $x=1$, stop so for example, starting from 15, the list will be: 15, 46, 23, 70, 45, ...
0
votes
1answer
81 views

Proofs for the sequence $a_n$ with $a_n = \cos\left(\frac{n\pi}{4}\right)$

Proves for the sequence $a_n$ with $a_n = \cos\left(\frac{n\pi}{4}\right)$ a)Show that it exist an $m \in \mathbb{N}$ that $a_n=a_{n+m} \forall n \in \mathbb{N}$ and determine $M=\{a_n ; n \in ...
1
vote
3answers
182 views

Convergence of parameterized series $\sum_{n=2}^{+\infty} \left(( \sqrt{n+1} - \sqrt{n})^p \cdot \ln\left( \frac{n-1}{n+1}\right) \right)$

$$\sum_{n=2}^{+\infty} \left(( \sqrt{n+1} - \sqrt{n})^p \cdot \ln\left( \frac{n-1}{n+1}\right) \right)$$ I guess that more useful form is: $$\sum_{n=2}^{+\infty} \left(( \sqrt{n+1} + \sqrt{n})^{-p} ...
2
votes
1answer
185 views

Do there exist sequences with these properties and these limit points?

Prove or disprove the following statements of sequences: There is a bounded sequence ${a_n}$ with three limit points -8, 22 and 23. There is an unbounded sequence ${a_n}$ with three limit ...
1
vote
3answers
194 views

The rearrangement theorem for improper convergent series

The rearrangement theorem in real analysis tells us that given an absolutely convergent series we can permute its entries and the resulting series will have the same limit. But what happens if my ...
2
votes
1answer
77 views

How to show that $\lim_{n\to \infty}n\sin(2\pi en!)=2\pi$?

Please give me some hint to proceed. I'm clueless: Show that, $\lim_{n\to \infty}n\sin(2\pi en!)=2\pi$
0
votes
1answer
246 views

Is any convergent subsequence of a bounded sequence of L2 functions uniformly convergent

I think this is a relatively straight-forward question. It is part of a larger proof that I am trying to do. If I have an arbitrary, bounded sequence of functions in $L^2(\mathbb{R}^n)$, is any ...
0
votes
1answer
339 views

Sum of an infinite series with regularized gamma functions

I'm wondering if the following series has a closed form: $$S = e^{-x} \sum_{k=1}^{\infty} \left[\frac{x^k}{k!} \cdot \left(e^{-y} \sum_{l=0}^{k-1} \frac{y^l}{l!} \right) \right]$$ I occasionally ...
1
vote
1answer
100 views

Show That This Complex Sum Converges

For complex $z$, show that the sum $$\sum_{n = 1}^{\infty} \frac{z^{n - 1}}{(1 - z^n)(1 - z^{n + 1})}$$ converges to $\frac{1}{(1 - z)^2}$ for $|z| < 1$ and $\frac{1}{z(1 - z)^2}$ for $|z| > ...
3
votes
3answers
124 views

Find the sum of series

Does there exist an explicit formula for the sum of the series $$ \sum_{n=1}^\infty \frac{1}{n^2-z^2}? $$
5
votes
3answers
119 views

Use of Recursively Defined Functions

Recursion is definitely fascinating and can generate sequences that would need lengthy functions. While doing combinatorics, I found that certain counting problems and some probability computation ...
3
votes
1answer
434 views

comparison test of $\sum\frac{\sqrt{n+1}-\sqrt{n}}{n}$

I want to know if the sum $$\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$$ converges or not. I've tried the ratio and root test but they don't fit. So wolframalpha says that the sum converges by the ...
2
votes
0answers
105 views

$\zeta(2)$ and $\zeta(3)$ [duplicate]

Possible Duplicate: Riemann zeta function at odd positive integers We know that $\zeta(2)=\frac{\pi^2}{6}$ and also for every even number it's known, but it's still unknown what the exact ...
0
votes
1answer
76 views

Maps sending weakly convergent sequences to weakly convergent sequences are continuous?

Well, the question is in the title. I understand that they are continuous in the weak topology, but can't see that it must hold for the norm topology. Please help me.
2
votes
2answers
205 views

Finding sums of infinite convergent series

All calculus textbooks I know of seem to be obsessed with the question of which infinite series is convergent and which is not but none address the question of how to find the sum of an infinite ...
4
votes
1answer
78 views

A reference for some fact in analysis

I am looking for a reference for the following fact. Any hints would be appreciated. Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
2
votes
4answers
77 views

How can I prove that the value of this series is smaller than a constant c < 1

I am stuck in a proof in graph theory (random graphs) because in the last part, I need to prove the following: $$ \sum_{k=3}^n \frac{1}{4^k e^{2k}} < c $$ for some constant $c < 1$ and any ...
4
votes
2answers
357 views

Testing Convergence of $\sum \sqrt{\ln{n}\cdot e^{-\sqrt{n}}}$

What test should i apply for testing the convergence/divergence of $$\sum_{n=1}^{\infty} \sqrt{\ln{n}\cdot e^{-\sqrt{n}}}$$ Help with hints will be appreciated. Thanks
1
vote
1answer
73 views

Confused as to the right answer to this summation, am I wrong (most likely) or is the answer provided wrong?

If you have $\sum_{n = 0}^\infty(4/5)^n$ and you are asked to represent it as a geometric series you would: $\sum_{n = 0}^\infty(4/5)(4/5)^{n-1}$ //factor out your constant therefore $a = 4/5$, ...
1
vote
4answers
266 views

To prove an Arithmetic Progression

If the $p^{th}$ term of an arithmetic progression is $\alpha$ and the $q^{th}$ term is $\beta$, prove that the sum of its $p+q$ term is ...
1
vote
2answers
68 views

What formula could I use to represent this series?

Suppose I have following series - $$1, 2, 4, 7, 11, 16, \dots$$ How can I mathematically represent this series? I can't represent it as AP as d is not constant. ...
2
votes
3answers
156 views

Finite sequences of unbounded value

If I have a finite sequence of expressions $a_1+a_2+a_3+....a_k=\infty$, does that imply that at least one such $a_j=\infty$? I know that if it didn't it would make the sum not equal to infinity, but ...
6
votes
3answers
182 views

Help with infinite sum

Can you guys give me a hint on evaluating $$\sum_{n=1}^\infty \frac{1}{n(n+2)(n+4)}?$$ I have tried partial fractions but the series is not telescopic (at least I cannot see it)...
2
votes
3answers
280 views

Simple inequality involving exponential function

Can anyone explain why, $$\displaystyle 1 - e^{-\frac{1}{k}} \geq \frac{1}{ke}$$ for $k \geq 1$. This inequality is used to show that the series $$\sum_{k=1}^\infty 1 - e^{-\frac{1}{k}} $$ is ...
2
votes
2answers
517 views

Time series and social network analysis

I am interested about plotting graphs of a phenomenon and study it using tools from social network analysis. Suppose the nodes are time series, and that the links between the nodes are the correlation ...
3
votes
1answer
157 views

Vonmangoldt sums

The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $\ln(p)$ when it is a prime power say, $n=p^j$, is ...
1
vote
0answers
60 views

Matrix Representation of Integer Series

I would like some feedback regarding this process or the meaning of this process. Let say that I have a discrete time series: S = [1 2 3 4 5] And that I represent this serie by a stochastic matrix M ...
3
votes
1answer
236 views

Convergence\Divergence of $\sum_{k=2}^{\infty} \frac{\cos(\ln(\ln k))}{\ln k}$

Test for convergence the series $$\sum_{k=2}^{\infty} \frac{\cos(\ln(\ln k))}{\ln k}$$ My first thought was related to the use of the integral test, but things seem hard. Could we resort here to some ...
0
votes
3answers
51 views

how to find a $f(x)$ equal the given series

look at this series: $$\sum\limits_{n = 1}^\infty {\dfrac{{{{( - 1)}^{n - 1}}}} {{2n - 1}}{x^{2n}}}$$ by Cauchy-Hadamard formula,the above series convergence region is $(-1,1)$. at the end points, ...
4
votes
2answers
144 views

Show that a sequence converges

Suppose that $\{ x_n \}_{ n = 1 }^\infty$ is a converging sequence of real numbers with $\lim_{ n \to \infty } x_n = x$. Define $$ y_n = \frac{ x_1 + \cdots + x_n }{ n } $$ Show that $\lim_{ n \to ...
4
votes
5answers
85 views

Help showing a sequence is convergent

Suppose that a sequence $y_n$ is defined iteratively by $y_0 = 1$ and $$ y_{ n + 1 } = \frac{ 1 }{ 2 + y_n } $$ Show that $\{ y_n \}_{ n \geq 0 }$ is a convergent sequence. I'm not really sure where ...
4
votes
2answers
355 views

How to prove that construction of Farey sequence by mediant is coverage?

Farey sequence of order $n+1$ ($F_{n+1}$) can be construct by adding mediant value (${a+c \over b+d}$) into $F_{n}$, where ${a \over b}$ and ${c \over d}$ are consecutive term in $F_{n}$, and $b+d = ...
0
votes
1answer
41 views

Progressions with variable density that can be described in constant space?

Say we have an arithmetic progression in $Z_n$ like $3, 6, 9, 12, ...$ etc. If you move a sliding window of at least 3 values over the progression the 'density' in that subset compared to if the ...
0
votes
1answer
161 views

Yes or No, Real Analysis, continuity, compactness

Am I correct over statements below? The limsup and liminf of the sequence $n^2$ (meaning $1,4,9,16,\dots$) are equal. T Every bounded sequence has at most one ...
4
votes
2answers
626 views

proving cauchy condensation test

I have to prove the condensation test of cauchy by tomorrow and I am really unconfident about what I did: $$\sum_{n=1}^\infty a_n\text{ converges }\Leftrightarrow\sum_{n=1}^\infty 2^na_{2^n}\text{ ...
0
votes
2answers
119 views

Question on fixed point

I had some trouble to approach the question above. Especially (2) and (3). I appreciate if you can help!
0
votes
2answers
77 views

convergent but not absolutely convergence series problem which has zero sum

It is given that the series $\sum_{n=1}^\infty a_n$ is convergent, but not absolutely and $\sum_{n=1}^\infty a_n=0$. Denote by $S_k$ the partial sum $\sum_{n=1}^k a_n$ , $k=1,2,\dots$ Then, (a) ...
1
vote
1answer
395 views

Proof about diameter of a set

I could not prove the following question could you please help me? Best Regards Let $X, d(x, y)$ be a metric space. By definition, diameter of a bounded set $A ⊂ X$ is the number $diam(A)$ = ...
0
votes
4answers
344 views

For which x values, this series is convergent?

Find all real number $x$ such that the series: $$\sum_{n=1}^\infty {n x^n\over 2n^2+1}$$ is absolutely convergent?
2
votes
2answers
137 views

Every sequence is composed of isolated points?

Let $(M,d)$ be a metric space and $\{x_n\}_{n=1}^\infty\subset M$ be a sequence. Prove that $$\forall n\in\mathbb N,\quad\exists \varepsilon> 0 \;B(x_n,\varepsilon)\cap \{x_n\}_{n=1}^\infty = ...
0
votes
2answers
226 views

Infinite sum of product of sequences convergent?

Let $(a_n)_{n\in \mathbb N}$ be a sequence of real numbers with $a_n\ge0$ for all $n\in\mathbb N$ and suppose $\sum_{n=1}^\infty a_n$ converges. Show that if $(b_n)_{n\in\mathbb N}$ is any bounded ...
3
votes
3answers
167 views

Series of positive terms

I want to show that $\displaystyle 1+\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+\cdots$ converges. I know that by using D'Alembert ratio test I easily show that this series converges but I am doing in ...
2
votes
1answer
63 views

Finding the minimizing vector of a $l_{2}$ sequence

I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated. Let $X = l_{2}$. Let $x \in X$ be given by $x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$ Let $M ...
3
votes
2answers
49 views

An absolute convergence criterion in $\Bbb C$

Here's a problem I'm having trouble with: Show that if $\sum u_k$ converges for $u_k\in\Bbb C$, and $|\arg(u_k)|\leq c<\pi/2$ for all $k$, then $\sum |u_k|$ converges too. All I have after ...
1
vote
2answers
93 views

The convergence of the sequence $x_{n+1}= \frac{n}{n+1} x_n$

I wanted to examine the convergence of the series $$\displaystyle x_{n+1}= \frac{n}{n+1} x_n$$ and try to find its limit, but I'm having difficulty doing so. The only test I thought would be useful ...