Convergence of sequences and different modes of convergence.

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42
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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 ...
3
votes
6answers
233 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, $u_{n}=\frac{1}{\sqrt{n}}$ ...
5
votes
2answers
2k 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? ...
8
votes
3answers
449 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?
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 ...
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 ...
4
votes
3answers
2k views

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

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 ...
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) ...
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 ...
5
votes
3answers
693 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 ...
2
votes
2answers
184 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ ...
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 + ...
21
votes
2answers
737 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. ...
7
votes
5answers
580 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 ...
2
votes
2answers
373 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 ...
14
votes
1answer
679 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.) ...
8
votes
2answers
379 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 ...
12
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5answers
657 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} ...
10
votes
1answer
379 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 ...
5
votes
1answer
2k 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 ...
9
votes
2answers
1k views

Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?

Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges. I calculated $$ ...
9
votes
2answers
1k 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 ...
5
votes
4answers
124 views

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

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.
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 ...
3
votes
1answer
275 views

Prove: If $\sum \limits_{n=1}^{\infty}|f_n|$ converges uniformly, so does $\sum \limits_{n=1}^{\infty}f_n$.

I'd like your help proving that If $\sum\limits_{n=1}^{\infty}|f_n|$ converges uniformly, so does $ \sum\limits_{n=1}^{\infty}f_n$. There a Weierstrass theorem saying that if there's a positive ...
7
votes
1answer
383 views

Uniform convergence of infinite series

Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
3
votes
1answer
230 views

Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$

Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$ for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
2
votes
2answers
153 views

Proof for convergence of a given progression $a_n := n^n / n!$

"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable. (a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$ [...]" I am having problems ...
1
vote
1answer
115 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
5
votes
7answers
2k 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 ...
8
votes
2answers
2k views

When can the order of limit and integral be exchanged?

I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of taking limit wrt one ...
8
votes
1answer
429 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...
6
votes
3answers
458 views

Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$

I'd like your help with the following claim to prove: $$\lim_{n \to \infty} \int_{0}^{\sqrt n}\left(1-\frac{x^2}{n}\right)^ndx=\int_{0}^{\infty} e^{-x^2}dx.$$ I think I should use the claim: Let ...
6
votes
3answers
2k views

Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
5
votes
4answers
298 views

Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$

Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$? I've calculated that the recurrence relation for this integral is: $\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
5
votes
2answers
1k 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 ...
4
votes
1answer
197 views

Convergence of $\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}$

Im interested to know for which $\alpha \in \mathbb{R}$ the following integral converges: $$\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}.$$ In the answers to this post it was shown that the ...
3
votes
3answers
1k views

How to find the sum of the following series

How can I find the sum of the following series? $$ \sum_{n=0}^{+\infty}\frac{n^2}{2^n} $$ I know that it converges, and Wolfram Alpha tells me that its sum is 6 . Which technique should I use to ...
1
vote
2answers
140 views

Cluster Point in Fréchet-Urysohn Space is Limit Point?

A topological space is called a Fréchet-Urysohn space if for every $A \subset X$ and every $x \in \overline{A}$ there exists a sequence $x_1, x_2,....$ of points of $A$ converging to $x$. So, is it ...
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 ...
4
votes
2answers
118 views

Is Completeness intrinsic to a space?

Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric? If ...
4
votes
3answers
330 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
3
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3answers
115 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
3
votes
2answers
484 views

Converging series in Banach space

Does someone know if the following is true: Let $\mathbb{X}$ be some arbitrary Banach space. $\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$. Now, if the series ...
1
vote
5answers
139 views

Convergence Proof: $\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$

I have to check whether the following expression converges; if yes I have to give the limit. $$\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$$ Now I did the following: ...
1
vote
1answer
171 views

$X_n \stackrel{d}{\to} X$, $c_n \to c$ $\implies c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

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 $X_n \stackrel{d}{\to} X$. ...
21
votes
1answer
792 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 ...
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 ...
5
votes
4answers
2k views

Is the space $C[0,1]$ complete?

In order to prove $C[0,1]$ is complete, my functional analysis book says: "It is only necessary to show that every Cauchy sequence in $C[0,1]$ has a limit". It goes on by supposing $\{x_n\}$ is a ...
10
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 ...