Convergence of sequences and different modes of convergence.

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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 ...
20
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3answers
609 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 ...
18
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2answers
922 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 ...
12
votes
1answer
221 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 ...
10
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2answers
502 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 ...
9
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0answers
162 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
9
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1answer
316 views

Sum of cosines of primes

Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$ How to prove this series converges/diverges? $$\sum_{n=1}^\infty \cos{p_n}$$
8
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2answers
426 views

convergence of $\sum \limits_{n=1}^{\infty }\left\{ \dfrac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2}$

I am investigating the convergence of $$\begin{split}\sum _{n=1}^{\infty }\left\{ \dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2} &= \sum _{n=1}^{\infty ...
6
votes
2answers
469 views

Convergence/Divergence of a particular infinite nested radical

Is it known if the following infinite nested radical converges or diverges (for $n \in \mathbb N$)?: $$R(n) = \sqrt{n+\sqrt{(n+1)+\sqrt{(n+2)+ \cdots}}}$$ I recently became interested in these ...
6
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2answers
2k views

Dini's Theorem. Uniform convergence and Bolzano Weierstrass.

In Spivak's chapter on uniform convergence he asks to prove the following THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...
4
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1answer
386 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, ...
14
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1answer
336 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
14
votes
2answers
857 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
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1answer
486 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 ...
12
votes
2answers
325 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 ...
8
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2answers
536 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
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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
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0answers
102 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
4answers
183 views

Again, improper integrals involving $\ln(1+x^2)$

How can I check for which values of $\alpha $ this integral $$\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}\,dx $$ converges? I managed to do this in $0$ because I know $\ln(1+x)\sim x$ near 0. I ...
6
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3answers
325 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 ...
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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?
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1answer
45 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
103 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.
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2answers
107 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?
22
votes
0answers
973 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
597 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
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2answers
181 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, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
9
votes
3answers
2k 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
762 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
386 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
226 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
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4answers
120 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
504 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
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1answer
651 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
236 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
323 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 ...
4
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1answer
786 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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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) ...
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votes
1answer
842 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
267 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 ...
0
votes
1answer
183 views

Is this infinite series related to prime and composite numbers convergent?

I don't know whether this series converges: $$\frac{1}{4} - \frac{1}{5} + \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} - \frac{3}{11} + \frac{1}{12} - \frac{1}{13} + ...
9
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1answer
232 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
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2answers
225 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
146 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
243 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 $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
6
votes
7answers
197 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
277 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
371 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 ...