Convergence of sequences and different modes of convergence.

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$\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges? [duplicate]

Possible Duplicate: Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent If $\sum\limits_{n=1}^\infty |a_n|$ converges, the ...
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3answers
1k views

Which sequences converge in a cofinite topology and what is their limit?

This is an exercise from an earlier calculus 1 reading at my university: Let $X$ be a space containing infinitely many elements. In the cofinite topology, a set $\Omega$ is open iff $\Omega = ...
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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 ...
8
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2answers
440 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 ...
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1answer
284 views

Convergence in probability (limit of integrals)

How to prove the following: $$ \lim_{n \to \infty} \int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2+ \cdots +x_n^2}{x_1+x_2+ \cdots +x_n} dx_1 dx_2 \cdots dx_n = \frac23 $$ I would really ...
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237 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} ...
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1answer
495 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 ...
8
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1answer
831 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 ...
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442 views

If the sum and the product of two sequences converges to zero, does that mean that each sequence converges to zero?

If the sum and the product of two sequences converges to zero, does that mean that each sequence converges to zero ? Thanks
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2answers
240 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
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Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: ...
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2answers
538 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 ...
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1answer
338 views

Does the integral test work on higher dimensions?

The integral test of convergence states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty ...
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2answers
222 views

How will the limit $\lim \sin(A^{n})$ behave for $|A| > 1$?

While answering to the following question Problem. Find the range of $x$ for which $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \sin(x^{-n})$ converges. Unlike some careless answerers who merely ...
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4answers
90 views

$a_{n+f(n)}-a_{n}\rightarrow0$ implies convergence?

$(a_{n})$ is a sequence of reals. Say $a_{n+f(n)}-a_{n}$ tends to 0 as n tends to infinity for every function f from the positive integers to the positive integers. Does this imply that $a_{n}$ is ...
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1answer
404 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 ...
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1answer
320 views

Convergence in $L^1$ space

Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$. Prove that ...
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135 views

Determine if $\sum \frac{1}{(\ln{n})^{\ln{n}}}$ converges.

I currently trying to determine it by comparison. I've tried comparing it with $\frac{1}{n^2}$, and it seems to work but I'm not sure if I did it right. I've done it like this: $(\ln ...
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2answers
227 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 ...
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+100

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
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5answers
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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 ...
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755 views

Test for convergence $\sum_{n=1}^{\infty} \frac{1}{2^\sqrt{n}}$ [duplicate]

Possible Duplicate: convergence of a series involving $x^\sqrt{n}$ Test for convergence $$\sum_{n=1}^{\infty} \frac{1}{2^\sqrt{n}}$$ My first thought was to use the ratio test but it's ...
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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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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. ...
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153 views

the sum $\sum \limits_{n>1} f(n)/n$ over primes

Let $$ f(n)=\begin{cases}-1&\text{if $n$ is a prime integer},\\ 1&\text{otherwise}. \end{cases} $$ Then, does the series $$ \sum_{n>1} f(n)/n $$ converge or diverge?
7
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5answers
495 views

does $\int_0^\infty x/(1+x^2 \sin^2x) \mathrm dx$ converge or diverge?

$$\int_0^\infty x/(1+x^2\sin^2x) \mathrm dx$$ I'd be very happy if someone could help me out and tell me, whether the given integral converges or not (and why?). Thanks a lot.
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886 views

What is the radius of convergence of $\displaystyle\sum z^{n!}$?

How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
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194 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 ...
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3answers
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proving convergence of a sequence and then finding its limit

For every $n$ in $\mathbb{N}$, let: $$a_{n}=n\sum_{k=n}^{\infty }\frac{1}{k^{2}}$$ Show that the sequence $\left \{ a_{n} \right \}$ is convergent and then calculate its limit. To prove it is ...
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2answers
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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? ...
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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 ...
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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 ...
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147 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
7
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3answers
209 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) ...
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6answers
251 views

Series where the usual convergence tests fail

Is there any simple series where all the usual convergence tests are inconclusive? And how is convergence/divergence determined for these series?
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281 views

If $\frac{\sin{a_n}}{a_n}\rightarrow 1$ then $a_n\rightarrow0 $

I'm trying to prove that if $\;\frac{\sin{a_n}}{a_n}\rightarrow 1$ then $a_n\rightarrow0$, assuming $a_n\neq 0$ for all $n$. I think this is easy enough to show as follows: first, prove ...
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664 views

Uniform convergence problem

I encountered this problem while studying for an analysis exam. Here is a related question I asked some days ago. The problem is as follows: Suppose $a_n$ is a decreasing sequence of positive real ...
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2answers
146 views

for which value of $a$ that $ \big(\sum\frac {1}{u_n}\big) $ converges?

For any real number $a$ (positive or negative), define a sequence $\{u_n\}$ (depending on $a$) recursively by $u_0=2$ and $$ \int_{u_n}^{u_{n+1}}(\ln u)^a\,du=1.$$ For which $a\in\mathbb{R}$ does $$ ...
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125 views

Convergence of $\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$

Does the series $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$$ converge absolutely, converge conditionally, or diverge? I've tried applying the ratio test and the root test, and in both cases the ...
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299 views

If a fixed point is a limit of a subsequence of iterates, must the whole sequence converge to it?

Say $X$ is a compact metric space, with $f:X \to X$ continuous. Now, for any $x_0$, $f^n(x_0)$ must have a convergent subsequence, say $f^{n_i}(x_0) \to x_\infty$. If we know that any such limit is a ...
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215 views

How can I prove the convergence of a power-tower? [duplicate]

In here, I saw that $$x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$$ exists as a real number (convergent) if and only if $$x\in[e^{-e}, e^\frac{1}{e}].$$ How can I prove this??
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1answer
296 views

Convergence in the Box Topology.

Given the sequence in $\mathbb{R}^\omega$: $$y_1=(1,0,0,0\ldots), y_2=(\frac{1}{2},\frac{1}{2},0,0\ldots), y_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\ldots),\ldots$$ I know that it converges in the ...
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1answer
315 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
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188 views

methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$

As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
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1answer
249 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 $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
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1answer
605 views

Convergence of a product series with one divergent factor

I'm currently struggling with the following problem: Let $\displaystyle \sum_{k=1}^{\infty} a_k$ be a convergent series with $a_k \in \mathbb{R} \setminus \{0\}$. Then is there always a sequence ...
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2answers
467 views

Kummer's test - Calculus, Apostol, 10.16 #15.

I want to prove the following: Let $\{ a_n \}$ and $\{ b_n \}$ be two sequences with $a_n>0$ and $b_n>0$ for all $n \geq N$, and let $$c_n = b_n - \frac{a_{n+1}b_{n+1}}{a_n}$$ Then If there ...
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3answers
420 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 ...
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3answers
231 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
7
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1answer
126 views

convergence of series with $k!$

check if the following series converges: $\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$ where $k!!=k(k-2)(k-4)(k-6)...$ I came across this exercise while going trough some old exams. I'm ...