Convergence of sequences and different modes of convergence.
11
votes
1answer
641 views
Strong and weak convergence in $\ell^1$
Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
10
votes
5answers
440 views
Why does this process, when iterated, tend towards a certain number? (the golden ratio?)
Take any number $x$ (edit: x should be positive, heh)
Add 1 to it $x+1$
Find its reciprocal $1/(x+1)$
Repeat from 2
So, taking $x = 1$ to start:
1
2 (the + 1)
0.5 (the reciprocal)
1.5 (the + 1)
...
5
votes
2answers
613 views
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? ...
8
votes
3answers
277 views
Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
1
vote
2answers
169 views
prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:
$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$
$Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$
$p=\int_{-1}^{1}Z(x)dx$
...
5
votes
3answers
545 views
Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the ...
30
votes
3answers
815 views
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
14
votes
1answer
429 views
Examples of Taylor series with interesting convergence along the boundary of convergence?
In most standard examples of power series, the question of convergence along the boundary of convergence has one of several "simple" answers. (I am considering power series of a complex variable.)
...
13
votes
5answers
479 views
Slowing down divergence 2
Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that
$
\sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)}
$
diverges.
(Why) must the series $$\sum_{n>1} ...
9
votes
2answers
835 views
Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $[0,2\pi]$
This question arises because of a problem I was doing (Bartle 3rd edition, section 9.4 problem 3). It was like this. Given $a_n$ a decreasing sequence of positive numbers and suppose that ...
10
votes
3answers
608 views
Product of two power series
Say if I define a power series over some arbitrary field $F$ as
$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$
Then can I say:
$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
6
votes
1answer
264 views
Uniform convergence of infinite series
Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
15
votes
1answer
901 views
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this
$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$
$$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$
where $m$ is an ...
5
votes
7answers
1k views
Series that converge to $\pi$ quickly
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
6
votes
2answers
1k views
When can the order of limit and integral be exchanged?
I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of
taking limit wrt one ...
4
votes
1answer
163 views
Convergence of $\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}$
Im interested to know for which $\alpha \in \mathbb{R}$ the following integral converges:
$$\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}.$$
In the answers to this post it was shown that the ...
3
votes
3answers
656 views
How to find the sum of the following series
How can I find the sum of the following series?
$$
\sum_{n=0}^{+\infty}\frac{n^2}{2^n}
$$
I know that it converges, and Wolfram Alpha tells me that its sum is 6 .
Which technique should I use to ...
17
votes
7answers
967 views
“Why do I always get 1 when I keep hitting the square root button on my calculator?”
I asked myself this question when I was a young boy playing around with the calculator.
Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman ...
4
votes
2answers
102 views
Is Completeness intrinsic to a space?
Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric?
If ...
4
votes
3answers
225 views
Convergence of a sequence of periodic functions
Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question.
Let the ...
2
votes
1answer
153 views
Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
21
votes
1answer
633 views
How to prove convergence of polynomials in $e$ (Euler's number)
These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$
This function goes to 2. I've calculated this with sage math ...
10
votes
1answer
911 views
Uniform convergence of series $\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$
Using Dirichlet series test I've proves that series $\displaystyle\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$ converges for all $x\in\mathbb{R}$.
How to determine whether the series ...
8
votes
5answers
833 views
Is there a way of working with the Zariski topology in terms of convergence/limits?
As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...
4
votes
4answers
1k views
Is the space $C[0,1]$ complete?
In order to prove $C[0,1]$ is complete, my functional analysis book says:
"It is only necessary to show that every Cauchy sequence in $C[0,1]$ has a limit".
It goes on by supposing $\{x_n\}$ is a ...
10
votes
1answer
165 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
4
votes
6answers
557 views
Fastest Square Root Algorithm
What is the fastest algorithm for finding the square root of a number?
I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
13
votes
1answer
549 views
Infinite tetration, convergence radius
I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework.
The problem is stated as follows. Assume we have an infinite ...
9
votes
1answer
180 views
Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$
converges?
I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
6
votes
3answers
1k views
Examples of function sequences in C[0,1] that are Cauchy but not convergent
To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
5
votes
4answers
476 views
Convergence or divergence of $\sum_{k=1}^{\infty} \left(1-\cos\frac{1}{k}\right)$
Does $$\sum_{k=1}^{\infty} \left(1-\cos\frac{1}{k}\right)$$ converge or diverge?
2
votes
1answer
114 views
Conditional expectation and martingales
I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$.
We ...
7
votes
2answers
585 views
Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?
Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges.
I calculated
$$
...
6
votes
3answers
381 views
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
I'd like your help with the following claim to prove:
$$\lim_{n \to \infty} \int_{0}^{\sqrt n}\left(1-\frac{x^2}{n}\right)^ndx=\int_{0}^{\infty} e^{-x^2}dx.$$
I think I should use the claim:
Let ...
6
votes
4answers
326 views
Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$
Please help me to prove that $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$ converge and it's limit $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: ...
6
votes
4answers
324 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.
...
4
votes
3answers
887 views
Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$
I need to check if
$$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
3
votes
1answer
216 views
Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$
Show that if a power-series converges for any value of $z_{0}$ of $z$, it will be absolutely convergent for all values of $z$ whose representation points are within a circle which passes through ...
2
votes
2answers
272 views
Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]
Possible Duplicate:
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
I need some help calculating the above limit.
What i have ...
2
votes
2answers
185 views
Prove that the sequence converges
Prove that the sequence converges.
For each positive integer $n$, let
$$y_n = 1 + \frac12 + \frac13 + \cdots + \frac1n - \int_1^n \frac{dx}x.$$
1
vote
3answers
202 views
How to calculate $\sum_{n=0}^\infty {(n+2)}x^{n}$
I want to calculate the sum of $$\sum_{n=0}^\infty {(n+2)}x^{n}$$
I have tried to look for a known taylor/maclaurin series to maybe integrate or differentiate...but I did not find it :|
Thank you.
...
-1
votes
1answer
38 views
Alternating functional Series Convergence SOS…
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
5
votes
1answer
373 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 ...
4
votes
2answers
270 views
Absolute and uniform convergence of $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$
I am trying to show $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$ converges absolutely for all values of $z$ $(z=0$ excepted$)$, but does not converge uniformly near $z=0$.
I observed that ...
3
votes
6answers
112 views
Does the following series converge?
Does the following series converge ?
$$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{n}}+\ldots$$
Let, $u_{n}=\frac{1}{\sqrt{n}}$
...
2
votes
1answer
169 views
A problem about convergence…
I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
2
votes
3answers
227 views
$L^p$-space convergence
Let $({f_n})_{n\geq 1}$ be a sequence of functions on $[0,1]$ such that $\lim
_{n\rightarrow \infty}f_{n}(x)=f(x)$ almost everywhere with respect to Lebesgue measure and
$$\sup_{n\geq ...
2
votes
2answers
194 views
Convergence of Lebesgue integrals
I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong:
Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with ...
2
votes
2answers
118 views
Proof for convergence of a given progression $a_n := n^n / n!$
"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable.
(a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$
[...]"
I am having problems ...
1
vote
2answers
44 views
Must the sequence $X_n$ converge to $0$ in probability?
Let $X_1, X_2,\dots$ be a sequence of random variables with
$\lim_{n\to +\infty} E[|X_n|] = 0$.
Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
