Convergence of sequences and different modes of convergence.

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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
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1answer
334 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
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3answers
188 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
116 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
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3answers
459 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
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1answer
592 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
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2answers
148 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: ...
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1answer
295 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
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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 ...
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0answers
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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) ...
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1answer
646 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 ...
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1answer
253 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
224 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 ...
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2answers
211 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
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139 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 ...
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1answer
226 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
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3answers
381 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
7answers
184 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
243 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
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1answer
311 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 ...
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1answer
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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 ...
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1answer
51 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 ...
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1answer
175 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 ...
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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
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1answer
210 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
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1answer
248 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 ...
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2answers
786 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
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2answers
93 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
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2answers
219 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 ...
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2answers
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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 ...
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2answers
168 views

What does the following statement mean?

If $\mu(A) <\infty $, then from almost everywhere convergence follows the convergence in the measure . I don't understand what the "convergence in the measure" means. Waiting for your explanation. ...
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1answer
706 views

Uniform convergence, but no absolute uniform convergence

Can someone give an example of a series of functions $f_k(x)$ for which $\sum_{k=0}^{\infty} f_k(x)$ converges uniformly, and $\sum_{k=0}^{\infty} |f_k(x)|$ converges pointwise, but ...
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2answers
427 views

Does $M_n^{-1}$ converge for a series of growing matrices $M_n$?

$M_n$ is a $n\times n$ matrix with $M_{n+1}=\begin{pmatrix}M_n & a_n \\ b_n^T & c_n\end{pmatrix}$ and $a_n, b_n, c_n \to 0$ for $n\to \infty$. Is this sufficient to state $$ ...
3
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2answers
53 views

Finding the convergence

The sequence $a_n = \sqrt[n]{(3^n+5^n)}$ is convergent? Tried to resolve applying the the limit $n\to\infty$ but couldn't figure out how to finish it, any idea? Thanks
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1answer
53 views

What is the proper way to handle the limit with little-$o$?

I was hoping to show that $$\left(1-\frac{x}{n}+o\left(\frac{2x}{n}\right)\right)^n \xrightarrow{n\to\infty} e^{-x}$$ which would be just fine without the little-$o$. Trying binomial formula: ...
3
votes
2answers
91 views

Convergence of Roots for an analytic function

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.
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1answer
809 views

Is there a continuous positive function whose integral over $(0,\infty)$ converges but whose limit is not zero?

Is there a continuous positive function whose integral over $(0,\infty)$ converges but whose limit is not zero? closest I could think of is the Riemann or tent functions. but they are defined as $0$ ...
3
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1answer
493 views

Nested radicals

Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges. A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt ...
3
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3answers
583 views

How to prove that convergence is equivalent to pointwise convergence in $C[0,1]$ with the integral norm?

I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of ...
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3answers
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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?
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2answers
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What is $L^p$-convergence useful for?

Why do people care about $L^p$-convergence $f_n \rightarrow f$? Are there any interesting application of $L^p$-convergence? For example, if $p=\infty$, then the limit $f$ of the sequence $f_n$ of ...
2
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4answers
197 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
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2answers
119 views

$f_n \rightarrow 0$ a.e. on $[0,1]$ & $\int_{[0,1]} |f_n|^2 dm \leq 1$ $\implies$ $\int_{[0,1]} |f_n| dm \rightarrow 0$

Let $f_n : [0,1] \rightarrow \mathbf{R}$ be a sequence of measurable functions such that $\bullet$ $f_n \rightarrow 0$ a.e. on $[0,1]$. $\bullet$ $\int_{[0,1]} |f_n|^2 dm \leq 1$ for all $n \geq ...
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1answer
161 views

Rewrite sequence $\frac{x^{2^n}}{1-x^{2^n}}$ in the form $a_n - a_{n+p}$ where $p \in \mathbb{N}$

For arbitrary $x \in \mathbb{R} \setminus \lbrace -1, 1\rbrace$, how can one rewrite the sequence $\frac{x^{2^n}}{1-x^{2^n}}$ in the form $a_n - a_{n+p}$ where $p \in \mathbb{N}$? The background ...
2
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2answers
107 views

Weak Convergence of Positive Part

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $p\in (1,\infty)$. Suppose $u_n\in L^p(\Omega)$ is such that $u_n\rightharpoonup u$ in $L^p(\Omega)$. Define the positive part of $u$ by ...
2
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1answer
614 views

Stuck on proving uniform convergence

I am preparing for my analysis finals tomorrow and I have been stuck over two hours trying to solve the following problem: Let $f_1, f_2, \ldots$ be a convergent (pointwise) sequence of ...
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3answers
51 views

Prove $\frac{x}{1+n^2x^2}$ is uniformly convergent

I need to prove that \begin{equation}f_n(x)=\frac{x}{1+n^2x^2}\end{equation} Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
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1answer
53 views

Convergence of series with sum

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact convex set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i ...
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1answer
425 views

The coefficients in the inverse of unit lower triangular Toeplitz matrix

I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & ...
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4answers
3k views

Does the sequence $\frac{n!}{2^n}$ converge or diverge?

Does the following sequence $\{a_n\}$ converge or diverge? $$a_n=\dfrac{n!}{2^n}$$