Convergence of sequences and different modes of convergence.

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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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3answers
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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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5answers
1k 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 ...
8
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7answers
270 views

Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$

I have a sequence $(a_n)$ where for each natural number $n$, $$a_n = (1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$$ and I want to find its limit as $n\to\infty$. I obviously couldn't ...
8
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3answers
341 views

Convergence of $a_{n+1}=\frac{1}{1+a_n}$

We define $$a_{n+1}=\frac{1}{1+a_n}, a_0=c> 0$$ I stumbled across that sequence and Mathematica gives me that it converges against $\frac{1}{2} \left(\sqrt{5}-1\right)$ which doesn't even depend ...
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2answers
457 views

convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$

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 ...
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6answers
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Pointwise Convergence of $\sum \frac{\sin(\sqrt{n}x)}{n}$

I am having trouble in proving the pointwise convergence of $$ g(x)=\sum_{n=1}^\infty \frac{\sin(\sqrt{n}x)}{n}$$ for all real numbers $x$ using elementary methods (e.g. integral test, Weierstrass ...
8
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3answers
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$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Please help me check, if $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$ $f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
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223 views

$\sum\limits_{n=1}^\infty |a_n|$ 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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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 ...
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3answers
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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
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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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2answers
456 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 ...
8
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1answer
297 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 ...
8
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1answer
547 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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2answers
230 views

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n(\sin(n)+2)}$ converge or diverge?

I was thinking about it and was stumped. Mathematica claims it converges.
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4answers
510 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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3answers
288 views

Does the series $1 + \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots$ converge?

Does the following variant of the harmonic series converge? If it diverges (which I think it does), can I know if it diverges to $\infty$ or has no limit? Note that the series is not alternating in ...
8
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2answers
250 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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2answers
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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: ...
8
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2answers
583 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 ...
8
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1answer
348 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 ...
8
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1answer
269 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 $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
8
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2answers
227 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 ...
8
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4answers
91 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 ...
8
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1answer
452 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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139 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 ...
8
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2answers
231 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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0answers
80 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
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0answers
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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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7answers
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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 ...
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Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
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5answers
839 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 ...
7
votes
3answers
147 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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2answers
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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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2answers
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
544 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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963 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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4answers
206 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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$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $ \lbrace a_{n}\rbrace $ be a sequence of positive terms such that $ \sum \limits_{n=1}^{\infty}a_{n} $ diverges. I am going to show that the series $$ \sum ...
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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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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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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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264 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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2answers
166 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, ...
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3answers
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 ...
7
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1answer
709 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 ...
7
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2answers
152 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 $$ ...
7
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1answer
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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 ...
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3answers
211 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) ...