Convergence of sequences and different modes of convergence.

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42
votes
11answers
4k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
31
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
24
votes
3answers
635 views

Convergence of a series with repeated sines

Show that the series $$ \sum_{n=1}^\infty \frac{\sin\big(\sin(n)\big)}{n}, $$ converges. More generally, show that for every $k\in\mathbb N$ the series $$ \sum_{n=1}^\infty \frac{\sigma_k(n)}{n}, $$ ...
21
votes
2answers
653 views

How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
21
votes
1answer
2k views

What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
21
votes
2answers
740 views

Find the value of the sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

How can we find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\big(1+\frac1n\big)^n\right\}$? This problem appears in: T. Andreescu, T. Radulescu & V. ...
21
votes
1answer
794 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
20
votes
1answer
1k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...
19
votes
7answers
1k views

“Why do I always get 1 when I keep hitting the square root button on my calculator?”

I asked myself this question when I was a young boy playing around with the calculator. Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman ...
19
votes
1answer
263 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} ...
18
votes
3answers
584 views

Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?

So we have$$\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}=x\\\sqrt{2}^x=x$$where $x=2$ heuristically seems like a good solution. However, $x=4$ seems like an equally good solution. I was told in passing ...
17
votes
3answers
445 views

Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational

Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
17
votes
4answers
390 views

A double sum $\sum \limits_{n=1}^{n=\infty}\left(\sum \limits_{k=n}^{k=n^2}\frac{1}{k^2}\right)$

How to evaluate $\displaystyle\sum_{n=1}^{n=\infty}\left(\sum_{k=n}^{k=n^2}\frac{1}{k^2}\right)$?
17
votes
6answers
316 views

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-…$

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$ The number of signs increases by one in each "block". I have an idea. Group the series like ...
17
votes
1answer
728 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...
16
votes
4answers
910 views

Why ${ \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent , but ${ \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent?

I don't understand why ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent, but ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent and its limit is equal to ...
16
votes
2answers
3k views

Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges?

We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges. Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the ...
16
votes
2answers
750 views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
15
votes
3answers
421 views

A question on convergence of series

Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$ $$ ...
14
votes
3answers
753 views

Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$

Prove convergence\divergence of the series: $$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$ Here is what I have at the moment: Method I My first way uses a result that ...
14
votes
5answers
526 views

Convergent or Divergent? $\sum_{n=1}^\infty\left(2^{\frac1{n}}-1\right)$

Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent? $$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$ I can't think of anything to compare it against. The integral looks too hard: ...
14
votes
1answer
684 views

Examples of Taylor series with interesting convergence along the boundary of convergence?

In most standard examples of power series, the question of convergence along the boundary of convergence has one of several "simple" answers. (I am considering power series of a complex variable.) ...
14
votes
2answers
728 views

If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?

This is a self-posed question, so I do not know the answer and I would like to know what do you think about. Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
14
votes
1answer
205 views

Does $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ converge for some $a\in(0,1)$?

The divergence of the series $\sum_{n=1}^\infty n^{-1-|\sin n|}$ is proved here. An inmediate consequence is that if $a\ge1$ then $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ also diverges. My question is: ...
14
votes
0answers
496 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
13
votes
2answers
1k views

A “non-trivial” example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it ...
13
votes
4answers
458 views

Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$

Evaluate the following limit: $$\lim \limits_{n\to \infty}\,\,\, n\!\! \int_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$ I have done the problem . How I solved is First I ...
13
votes
5answers
193 views

Show that the sequence ${a_n}$ converges where $a_n = \sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{n}}}}$ for $n\geq 1$.

The original question was to determine whether the sequence converges, but I have checked for extremely high values of $n$ and it seems as though it does converge. This lead me to wonder if there was ...
13
votes
1answer
435 views

A few counterexamples in the convergence of functions

I'm studying the various types of convergence for sequences of real valued functions defined on measure spaces: pointwise convergence a.e. , convergence in $L^p$ norm, weak convergence in $L^p$, and ...
13
votes
1answer
491 views

Series which are not Fourier Series

How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function? I have shown that the series is convergent by Dirichlet test. Let $a(n)=\frac{1}{\log ...
12
votes
2answers
311 views

Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?

Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
12
votes
5answers
658 views

Slowing down divergence 2

Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that $ \sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)} $ diverges. (Why) must the series $$\sum_{n>1} ...
12
votes
1answer
123 views

Does $f_n \to 0$ in $L^1(\mathbb R^2)$ imply that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?

I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise. Suppose you have $(f_n)_{n\ \in \mathbb N}\subset ...
12
votes
2answers
182 views

Convergence of $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\binom{n}{k}\frac{\sin k}{k}$

Is this series convergent? I met it when I was studying some fractals. $$\lim_{n\to\infty}\sum_{k=1}^{n}\binom{n}{k}\frac{\sin k}{k}$$
12
votes
1answer
332 views

Convergence of the series $\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $

Study the convergence of the next series: $$\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $$ My solution: since $$\frac{(2n)!!}{(2n+2)!!} \leq \frac{(2n)!!}{(2n+1)!!}$$ forall $n \in \mathbb{N}$ and ...
12
votes
1answer
215 views

Showing a series is convergent. [duplicate]

Possible Duplicate: Contest problem about convergent series Let ${p}_{n}\in \mathbb{R} $ be positive for every $n$ and $\sum_{n=1}^{∞}\cfrac{1}{{p}_{n}}$ converges, How do I show that ...
11
votes
5answers
1k views

Is there a way of working with the Zariski topology in terms of convergence/limits?

As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...
11
votes
3answers
222 views

Find the limit of $(\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot … \cdot \sin 1)^{\frac{1}{n}}$

Could you tell me how to find $\lim_{n \rightarrow \infty} (\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot ... \cdot \sin 1)^{\frac{1}{n}}$ ?
11
votes
3answers
271 views

Divergence for $p$ prime numbers and convergence for $m$ composite numbers

Does there exist a sequence $(a_n)_{n\in \mathbb N} \in \mathbb C^{\mathbb N}$ such that : For all $p$ prime numbers the series $\displaystyle \sum_{n\in \mathbb{N}} a_n^p$ diverges, and for ...
11
votes
3answers
1k views

Product of two power series

Say if I define a power series over some arbitrary field $F$ as $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$ Then can I say: $$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
11
votes
2answers
287 views

Generalized Fibonacci Sequence Question

The Fibonacci Sequence is defined as the recurrence $a(n)=a_{n-1}+a_{n-2}$ where $a(0)=0$ and $a(1)=1$. Today, I was bored so I considered the sequence $a(n)=\sqrt{a_{n-1}}+\sqrt{a_{n-2}}$. Ten ...
11
votes
1answer
1k views

Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
11
votes
2answers
169 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
11
votes
1answer
91 views

Existence of a specific reordering bijection

Please consider a bijection $g:\mathbb{N}\rightarrow\mathbb{N}$ with following properties: For all real series $(a_n)_{n\geq1}$, convergence of $\sum_{n=1}^{\infty}a_n$ implies convergence of ...
11
votes
1answer
183 views

$L^{2}(\mathbb R)$- norm of entire function

Let $f(z)$ be an entire function defined by $$f(z)=\prod_{n=1}^{\infty}\bigg(1-\frac{z^{2}}{a_{n}^{2}}\bigg),\qquad z\in \mathbb C$$ where $\{a_{n}\}_{n=1}^{\infty}$ is a sequence of positive real ...
10
votes
8answers
534 views

Proving convergence of a sequence whose terms are integrals

How to prove the following sequence converges to $0.5$ ? $$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$ What I have tried: I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n ...
10
votes
5answers
496 views

Why does this process, when iterated, tend towards a certain number? (the golden ratio?)

Take any number $x$ (edit: x should be positive, heh) Add 1 to it $x+1$ Find its reciprocal $1/(x+1)$ Repeat from 2 So, taking $x = 1$ to start: 1 2 (the + 1) 0.5 (the reciprocal) 1.5 (the + 1) ...
10
votes
2answers
408 views

Prove $\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty$ for an increasing sequence $a_n$ of positive integers

The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that: $$ \sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty $$ While I have ...
10
votes
1answer
384 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converges? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
10
votes
3answers
591 views

Does $f_{n}(x)=n\cos^n x \sin x$ uniformly converge for $x \in [0,\frac{\pi}{2}]$?

I want to check whether the following function is uniformly converges: $f_n(x)=n\cos^nx\sin x$ for $x \in \left[0,\frac{\pi}{2} \right]$. I proved that the $\lim \limits_{n \to \infty}f_{n}(x)=0$ for ...