Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

8
votes
1answer
333 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
2answers
132 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
votes
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 ...
8
votes
0answers
490 views

Difference of differentiation under integral sign between Lebesgue and Riemann

Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
7
votes
5answers
668 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 ...
7
votes
5answers
640 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
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 ...
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. ...
7
votes
2answers
152 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
votes
2answers
806 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 ...
7
votes
5answers
423 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.
7
votes
3answers
968 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 = ...
7
votes
6answers
238 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?
7
votes
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
votes
2answers
131 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
votes
2answers
122 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 ...
7
votes
3answers
256 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 ...
7
votes
1answer
200 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 ...
7
votes
1answer
284 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 ...
7
votes
1answer
166 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 ...
7
votes
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
votes
1answer
556 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 ...
7
votes
1answer
318 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 ...
7
votes
2answers
344 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 ...
7
votes
3answers
380 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 ...
7
votes
3answers
219 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
votes
2answers
200 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 ...
7
votes
1answer
124 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 ...
7
votes
1answer
184 views

What functions satisfy $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ^{\infty} f(x_n) < \infty $ [duplicate]

Could you help me with this problem? What functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following implication? $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ...
7
votes
1answer
633 views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
7
votes
1answer
144 views

Question dealing with a series of functions that uniformly converges on $[0,1]$

Let $f_1, f_2, \ldots$ be a sequence of continuous positive functions of $[0,1]$ and let $a_n = \sup\{ f_n(x) : x \in [0,1]\}$. In class, we showed that if $\sum f_n$ uniformly converges on $[0,1]$, ...
7
votes
1answer
125 views

$\Gamma(1/2-n+it)$ converges uniformly

Prove that $\Gamma(1/2-n+it)\rightarrow 0$ uniformly as $n\rightarrow\infty$ for $t\in\mathbb{R}$, where $n$ is a positive integer. I'm not sure which definition of $\Gamma$ would be easiest to ...
7
votes
2answers
100 views

Convergence of $\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$

If $\sum_{n=2}^{\infty}a_n$ converge, then also converge this series? $$\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$$ Please verify my answer below Counterexample: ...
7
votes
1answer
400 views

Uniform convergence of infinite series

Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
7
votes
1answer
135 views

Zeros in the complex plane and convergence

I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty. A text I am reading states the following: ...and given that ...
7
votes
2answers
131 views

Series convergence/divergence

I was trying to prove the following question. Part a is intuitive but couldn't give a clear mathematical argument. For parts b and c It seems there is something I am not seeing. Any help ? If ...
6
votes
4answers
372 views

What steps should I be doing to determine if this series is convergent or divergent?

The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$ The first thing I did was use the divergence test which didn't help since the result of the limit was 0. If I multiply it through, the result ...
6
votes
4answers
456 views

What is the sense of the equality sign =?

Sometimes in real analysis, we write an equality and specify that is true in the sense of the $L^2$ norm. My question is, when in mathematics we write '=' and don't specify anything, what we really ...
6
votes
4answers
359 views

Power Series with the coefficients $n!/(n^n)$

I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series. Or, by Cauchy Hadamard, the limit of $(n!/(n^n))^{(1/n)}$ as n approaches infinity. ...
6
votes
7answers
2k views

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 ...
6
votes
6answers
3k views

Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
6
votes
3answers
334 views

Disprove uniform convergence of $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ in $[0,\infty)$

How would I show that $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ does not uniformly converge in $[0,\infty)$? I don't know how to approach this problem. Thank you.
6
votes
4answers
410 views

Convergence of Series

At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to ...
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
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
votes
4answers
2k views

proving convergence for a sequence defined recursively

The sequence $\left \{ a_{n} \right \}$ is defined by the following recurrence relation: $$a_{0}=1$$ and $$a_{n}=1+\frac{1}{1+a_{n-1}}$$ for all $n\geq 1$ Part 1)- Prove that $a_{n}\geq 1$ for all ...
6
votes
4answers
512 views

Convergence in metric and a.e

How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?
6
votes
5answers
301 views

Limit proof of a sqrt-heavy expression with binomial formula / sandwich-rule

Let be $$b_n := \sqrt{n+\sqrt{2n}}-\sqrt{n-\sqrt{2n}}, n\in\mathbb{N}$$ a sequence. I am to determine $\lim\limits_{n\to\infty}b_n$ which is obviously $\sqrt{2}$. My first step was to transform the ...
6
votes
3answers
2k views

Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
6
votes
4answers
686 views

Terms that get closer and closer together [duplicate]

Possible Duplicate: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy? I was thinking about sequences where it appears the terms get closer and ...