Convergence of sequences and different modes of convergence.

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14
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2answers
813 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 ...
13
votes
1answer
468 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 ...
8
votes
2answers
495 views

Prove that the convergence of the sequence ($s_n$) implies the convergence of ($s_n^3$)

I believe I have the gist of how to prove this. My professor worked out a problem similar to this one only, instead of ($s_n^3$), he used ($s_n^2$), and I am slightly confused as to how he came up ...
7
votes
2answers
2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
7
votes
2answers
1k views

Does $\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $ converge/diverge?

How would you prove convergence/divergence of the following series? $$\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $$ I'm interested in more ways of proving convergence/divergence for this series. Thanks. ...
6
votes
0answers
100 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
6
votes
3answers
293 views

Application of Central Limit Theorem

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
4
votes
1answer
319 views

Dominated convergence and $\sigma$-finiteness

I am curious about the Dominated Convergence Theorem for a sequence of functions that converges in measure. Theorem: Let $(X,\mathcal{S},\mu)$ be a measure space. If $\{f_n\}, f$ are measurable, ...
3
votes
1answer
1k views

Common Problems while showing Uniform Convergence of functions

All, I am having some trouble in checking whether a sequence of functions converges pointwise or uniformly. The problem is, sometimes, my intuition is right and sometimes its wrong. Finding the limit ...
2
votes
1answer
37 views

The interchange of limit for integration

This kind of problem bothers me for a while. Each time I meet such problem I got stuck and has to deal them case by case. So I post this problem here to ask for some general condition of the ...
2
votes
2answers
102 views

Find the Maclaurin series of f(x)

Find the Maclaurin series of $f(x)=\dfrac{1}{1+x+x^2}$ and the radius of convergence of the series. I can't solve this problem.
1
vote
2answers
105 views

trouble calculating sum of the series $ \sum\left(\frac{n^2}{2^n}\right) $

Find out the sum of the series $\displaystyle \sum\limits_{n=1}^{\infty} \dfrac{n^2}{ 2^n}$. I have checked the convergence, but how to calculate the sum?
21
votes
0answers
836 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
16
votes
1answer
583 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 ...
13
votes
2answers
175 views

Convergence of power towers

Let's define the sequence $\{s_n\}$ recursively as $$s_1=\sqrt2,\ \ \ s_{n+1}=\sqrt2^{\,s_n}.$$ Or, in other words, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
12
votes
2answers
322 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 ...
9
votes
3answers
1k views

In what spaces does the Bolzano-Weierstrass theorem hold?

The Bolzano-Weierstrass theorem says that every bounded sequence in $\Bbb R^n$ contains a convergent subsequence. The proof in Wikipedia evidently doesn't go through for an infinite-dimensional space, ...
8
votes
1answer
367 views

Sequence of convex functions

If a sequence of convex functions $\{ f_n \}$ on [0,1] converges pointwise to a continuous function f, then is the convergence uniform? The questions is almost identical to this one, except that the ...
8
votes
3answers
210 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
6
votes
4answers
118 views

Prove the limit for a series of products.

Let $\beta > 0$, $\lambda > 1$. Show the identity $$\sum_{n=0}^\infty\prod_{k=0}^{n} \frac{k+\beta}{\lambda + k + \beta} = \frac{\beta}{\lambda - 1}$$ I have checked the statement numerically. ...
6
votes
3answers
484 views

Proving that unconditional convergence is equivalent to absolute convergence

Regarding this discussion here: Absolute convergence a criterion for unconditional convergence. (thank you for the great answers, by the way) I'm still trying to do Exercise 3.2.2 (b) from these ...
6
votes
1answer
2k views

L'Hopital's rule and series convergence

I teach freshman calculus, and have recently been discussing series. In one question on a recent test, I asked whether $\sum\frac{n^2}{n^3+1}$ converges or diverges. One student got the correct ...
6
votes
1answer
612 views

Convergence of Integral Implies Uniform convergence of Equicontinuous Family

Let $\{f_n\}$ be an equicontinuous family of functions on $[0,1]$ such that each $f_n$ is pointwise bounded and $\int_{[a,b]} f_n(x)dx \rightarrow 0$ as $n\rightarrow \infty$, for every $ 0\leq a ...
5
votes
2answers
189 views

Does strong law of large numbers hold for an average of this triangular array containing “almost i.i.d.” sequences?

If a sequence of $n$ i.i.d. positive random variables $X_1,\ldots,X_n$ has the expectation $\mathbb{E}[X_1]=\mu<\infty$, then it is known that the strong law of large numbers (SLLN) holds: ...
5
votes
1answer
311 views

Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?

I need to prove the convergence/divergence of the series $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$ based on the convergence/divergence of the series $\sum_{n=1}^{\infty }a_{n}$. It is given that ...
5
votes
1answer
1k views

Which series converges the most slowly?

a_n converges more slowly then 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 for n=1 to n=inf. Ignoring constant factors, which type of function ...
3
votes
0answers
83 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
3
votes
3answers
3k views

Lebesgue Dominated Convergence example

How do I use Lebesgue Dominated Convergence Theorem to evaluate $$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt x}dx$$ What dominating function to use here?
2
votes
1answer
744 views

Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

It is stated in my professor's notes that, given a sequence $\{f_j\}$ of $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all ...
1
vote
1answer
261 views

Question based on Abel's theorem of multiplication of series

I am trying to show that the series $$\dfrac {1} {\sqrt {1}}-\dfrac {1} {\sqrt {2}}+\dfrac {1} {\sqrt {3}}-\ldots $$ is convergent, but that its square (formed by Abel's rule) $$\dfrac {1} {1}-\dfrac ...
9
votes
1answer
230 views

Convergence in topologies

Let $\tau_1\subseteq \tau_2$ be two Hausdorff regular topologies in an infinite set $X$ such that the convergences of sequences in $\tau_1$ and $\tau_2$ coincide (they have the same convergent ...
8
votes
2answers
219 views

Convergence in measure is not given by a seminorm

Let $V$ be the vector space of all real-valued Borel measurable functions on $[0,1]$. Show that convergence in measure (with respect to Lebesgue measure) is not given by a seminorm. That is, show that ...
7
votes
3answers
144 views

Evaluate $\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\cdots$

Evaluate $$ \frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{4}\cdots . $$ First, it is clear that terms tend to $1$. It seems that the infinity ...
7
votes
1answer
236 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
7
votes
3answers
394 views

Disproving uniform convergence

Given a sequence of functions $f_n$ which is known to be pointwise convergent how would you go about showing that is not uniform convergent? The particular example I'm working with is $f_n ...
6
votes
1answer
471 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
6
votes
7answers
190 views

Why $ \int^{+\infty}_{-\infty} x \, dx \neq 0 $

We are going over improper integrals and tests for convergence in my Calc II course. During a lecture, my professor warned us to take caution when taking integrals from negative infinity to infinity. ...
6
votes
3answers
254 views

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on ...
6
votes
1answer
343 views

If the partial sums of a $a_n$ are bounded, then $\sum{}_{n=1}^\infty a_n e^{-nt}$ converges for all $t > 0$

If the partial sums of a $a_n$ are bounded, then $$\sum_{n=1}^\infty \frac{a_n }{e^{nt}}$$ converges for all $t > 0$. Proof: since the partial sums of $a_n$ are bounded, then exists $C > 0 ...
5
votes
1answer
60 views

limit of sequence with factorial

How do you show that: $\lim\limits_{n\to \infty} \frac{\left(\frac{n}{2}\right)^{\frac{n}{2}}}{n!}=0$ using the squeeze theorem (I'd like to avoid using Stirling's formula, too). I tried rearranging ...
5
votes
2answers
312 views

Convergence of $\sum_n a_nb_n$ for all $b_n\searrow 0$ implies convergence of $\sum_n a_n$

I need a hint for a practice problem: Let $a_n \geq 0$. Show that if $\displaystyle\sum_{n=1}^\infty a_nb_n$ converges for every monotonically decreasing sequence $b_n \to 0$, then ...
5
votes
1answer
181 views

Showing a Lebesgue integral exists, while another doesn't.

Consider $$f_p(x)=x^p \exp\left(-x^8\sin^2x\right)$$ I have to show that $f_2\in\mathscr L(0,+\infty)$ whilst $f_3\notin \mathscr L(0,+\infty)$. Now, I am looking at the case $p=2$. The problematic ...
5
votes
0answers
74 views

Convergence of $\sum^n_{k=1}\frac1k$ after removing terms containing the digit $p$ [duplicate]

We know that $\sum^n_{k=1}\frac1k$ diverges. But if I were to pick a digit $p$ so that $p$ is an integer between $0$ and $9$ inclusive, and then I removed all terms in the sum $\sum^n_{k=1}\frac1k$ ...
5
votes
1answer
262 views

Convergence of nested radicals

In general if we have $$\sqrt{a_0+\sqrt{a_1+\sqrt{a_2....}}}$$ Are there easy ways of finding out if it converges, like from infinite products and series? Are there ways for converting the question ...
5
votes
1answer
255 views

expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$

This question relates to this answer I gave to a question about the integral $$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$ I derived an expansion in inverse powers of $p$ and then ...
5
votes
2answers
822 views

Convergence of $\Gamma(p)$ for $0<p\leq 1$ and divergence for $p \leq0$.

Can someone show me a proof or any clear resource about convergence of gamma function for values of $p$ less than zero. If possible I need proofs using integration by parts. My problem evaluating ...
4
votes
2answers
95 views

A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof

I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a ...
4
votes
2answers
234 views

Convergence of $\int_0^\infty \sin(t)/t^\gamma \mathrm{d}t$

For what values of $\gamma\geq 0$ does the improper integral $$\int_0^\infty \frac{\sin(t)}{t^\gamma} \mathrm{d}t$$ converge? In order to avoid two "critical points" $0$ and $+\infty$ I've ...
4
votes
3answers
202 views

Inequality regarding norms and weak-star convergence

Let $X$ be a normed space and $(x'_n) \subseteq X'$ a sequence of functionals where $x_n'$ has $x'$ has its limit in the *-weak topology in $X'$. Show that $$ ||x'|| \le \operatorname{lim inf}_{n\to ...
4
votes
2answers
2k views

Convergence in measure and almost everywhere

In a finite measure space, let $\{f_{n}\}$ be a sequence of measurable functions. Show that $f_{n} \rightarrow f$ in measure if and only if every subsequence $\{f_{n_{k}}\}$ contains a subsequence ...