Convergence of sequences and different modes of convergence.

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195
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17answers
14k 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 ...
41
votes
2answers
4k 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 ...
24
votes
6answers
20k views

Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $

I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,...
6
votes
3answers
4k views

Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$ [duplicate]

I need to check if $$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
12
votes
2answers
4k views

Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
11
votes
3answers
2k 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 ...
4
votes
6answers
368 views

Does the following series converge?

Does the following series converge ? $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{n}}+\ldots$$ Let, $u_{n}=\frac{1}{\sqrt{n}}$ $\lim\limits_{n\to\infty}{u_{n}}=\...
24
votes
2answers
2k views

Sum of reciprocals of numbers with certain terms omitted

I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \...
11
votes
1answer
4k views

Convergence of Ratio Test implies Convergence of the Root Test

In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise: Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\...
7
votes
6answers
968 views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
13
votes
5answers
865 views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
12
votes
2answers
8k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
6
votes
1answer
2k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
15
votes
1answer
3k 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 ...
10
votes
8answers
6k views

Series that converge to $\pi$ quickly

I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
16
votes
8answers
2k views

Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
27
votes
1answer
3k 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 ...
14
votes
2answers
796 views

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

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? 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 ...
11
votes
5answers
619 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) 0....
1
vote
1answer
687 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
43
votes
3answers
2k 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 + |\...
4
votes
1answer
631 views

Pointwise a.e. convergence and weak convergence in Lp

I'm trying to prove the following theorem: Let $\{f_n\}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \infty$) and $f_n ...
15
votes
4answers
839 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} \frac1{...
11
votes
3answers
3k views

Sufficient condition for convergence of a real sequence

Let $(x_n)$ be a sequence of real numbers. Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the ...
12
votes
1answer
2k views

Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...
8
votes
1answer
3k views

$L^1$ convergence gives a pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are http://terrytao.wordpress....
3
votes
2answers
133 views

usage of this condition

A very widely stated result: A sequence $x_n \to x$ iff every subsequence $x_{n^{\prime}}$ of $x_n$ contains a further subsequence $x_{n^{\prime\prime}}$ such that $x_{n^{\prime\prime}}\to x$. My ...
5
votes
2answers
423 views

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

Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$ Applying d'Alembert criterion I have that $$ \lim_{n\to\...
3
votes
1answer
403 views

If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
13
votes
1answer
455 views

Forcing series convergence

I am trying to figure this out: $\mathscr{S}=\big\{(a_n),(b_n),\dots \big\}$ is a finite set of real, null sequences. Does there exist a sequence $(\epsilon_n)$, where $\epsilon_k=\pm 1$ for each $...
17
votes
1answer
1k 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.) ...
7
votes
4answers
236 views

Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

How can we evaluate this series $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$ Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how ...
9
votes
2answers
3k views

Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $[0,2\pi]$

This question arises because of a problem I was doing (Bartle 3rd edition, section 9.4 problem 3). It was like this. Given $a_n$ a decreasing sequence of positive numbers and suppose that $$\sum_{n=0}^...
7
votes
3answers
476 views

Evaluate $ \sum\limits_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}$

The question was: Evaluate, ${\textstyle {\displaystyle \sum_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}}}.$ And I go, since $\frac{n}{n^{4}+n^{2}+1}\sim\frac{1}{n^{3}}$ and we know that ${\displaystyle \...
5
votes
2answers
300 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
3
votes
3answers
3k views

Proving the convergence of $\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$

I am attempting to prove if the below expression converges or diverges: $$\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$$ I decided to try using the limit convergence test. So then, I set $f(n) = \frac{...
8
votes
2answers
783 views

Cauchy nets in a metric space

Say that a net $a_i$ in a metric space is cauchy if for every $\epsilon > 0$ there exists $I$ such that for all $i, j \geq I$ one has $d(a_i,a_j) \leq \epsilon$. If the metric space is complete, ...
6
votes
4answers
193 views

Proof of $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$ [duplicate]

Why would $\lim_{n \to \infty} {a_n}^{1/n} = \lim_{n \to \infty}(a_{n+1}/a_n)$ be true where $(a_n)$ is a sequence in $\mathbb{R}$? Edit: Let all $a_n$ be positive.
4
votes
2answers
178 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
1
vote
2answers
1k views

For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ [duplicate]

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
2
votes
2answers
524 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
2
votes
1answer
290 views

Elementary proof, convergence of a linear combination of convergent series

Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also ...
-1
votes
1answer
97 views

The set of convergence of a sequence of measurable functions is measurable

Let $E$ be a measurable subset of $\mathbb{R}$ and let $F_n :E \to \mathbb{ R} $ be a sequence of measurable functions. Let $$E^*= \{ x\in E : F_n \text{ is convergent } \}$$ Is the set $E^*$ ...
21
votes
5answers
2k 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 ...
28
votes
2answers
1k views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$

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....
18
votes
3answers
4k 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 ...
11
votes
3answers
539 views

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in \...
8
votes
2answers
563 views

$\ell_1$ and unconditional convergence

Thanks to the Riemann theorem we know that absolute convergence and unconditional convergence are the same for $\mathbb{R}$. In all the Frechet spaces absolute convergence implies unconditional ...
6
votes
1answer
2k views

Which series converges the most slowly?

We say $a_n$ converges slower than $b_n$ if there exist an $x$ such that for all $m>x$, $a_m>b_m$ and both $\sum a_n$ and $\sum b_n$ converges. Ignoring constant factors, which type of function ...
1
vote
1answer
291 views

If $x_{m+n} \le x_n+x_m$, then $\lim x_n/n$ exists and is equal to $\inf x_n/n$

Let $(x_n)_{n \ge 1}$ be a sequence of real numbers satisfying $$x_{m+n} \le x_n+x_m$$ $m,n \ge 1$. Show that $\lim \limits_{n \to \infty} \dfrac{x_n}{n}$ exists and is equal to $\inf \left \{\...