Convergence of sequences and different modes of convergence.

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2answers
54 views

almost sure convergence given density [closed]

my problem: Let $X_n$ be iid random variables with density $f(x)=\frac{1}{2}x^{-2}1_{\{|x|>1\}}$. Show that $\frac{X_n}{n}$ does NOT converge almost surely. Can anybody help me?
3
votes
3answers
134 views

If $ \sum_{n=1}^{\infty}x_na_n $ converges when $x_n\to 0,$ then $ \sum_{n=1}^{\infty}a_n $ also converges. [duplicate]

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence with non negative terms so that for every sequence $\{x_n\}_{n=1}^{\infty}$ with $x_n \geq 0$ and $\lim_nx_n=0$, the series $\sum_{n=1}^{\infty}x_na_n$ ...
4
votes
1answer
77 views

Baby Rudin exercise 3.3 solution, possible typo in solutions manual?

Okay so I'm working through the exercises in Rudin and after checking my solutions manual for 3.3, I found something that seems like it can't be true. Here is the original question in rudin: ...
7
votes
2answers
192 views

Show that the series converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test ...
6
votes
1answer
82 views

Is there continuous $f: [0, 1] \rightarrow [0, \infty)$ such that for all $x$ there is $y$ with $f(y) < f(x)$?

I think there isn't. Here's a sketch of a proof. I'm just not sure whether it really works because I'm not confident with the transfinite versions of the standard theorems about limits and convergent ...
-1
votes
2answers
51 views

Why the length of the zigzag curve approximating the circle does not approach the length of the circle?

I recently bumped into this question which asks why $\pi=4$ is wrong. And some answers(see the answer of user TCL, for example) stated that this has to do with functions and their derivatives. ...
3
votes
2answers
74 views

Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} dx$.

Problem: Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} \, \mathrm{d}x$. I'm thinking about changing $\frac{1}{1-x}$ to $\sum x^k$ and then ...
2
votes
3answers
337 views

Why does this sum converge?

I know that the following sum converges to 2 via WolframAlpha, but I am not sure why. $$\sum_{k=1}^\infty k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = 2$$ WolframAlpha gives the ...
1
vote
0answers
27 views

Convergence of a hypergeometric function

The hypergeometric function, ${}_{2}F_1(a,b,c;z)$ can be written in terms of a power series in $z$ as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} ...
4
votes
2answers
115 views

Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior

What is the average value for $\mathrm{lcm}(a,b)$, with $ 1\le a \le b \le n$, for a given $n$, and what is the asymptotic behavior? The $\mathrm{lcm}$ is the least common multiple. I have ...
0
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0answers
21 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
1
vote
1answer
62 views

Dominated convergence theorem for complex-valued functions?

Suppose there is a sequence $\{f_n(x)\}$ such that $\lim_{n\rightarrow\infty}f_n(x)=f(x)$. I've previously used the dominated convergence theorem for interchanging the limit and the integral in ...
0
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1answer
50 views

Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has assymtoticaly an exponential distribution as $n \rightarrow \infty$

Let, $T_1,T_2 \cdots T_n $ be i.i.d random variables having reliability function: $R-(t) = 1 - \lambda t - o(t)$ as $t \rightarrow 0$. Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has ...
0
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0answers
93 views

Convergence of sum of a time series

After taking a course on SDEs I have started studying time series on my own. However, I am having difficulties in drawing parallelisms between the two subjects. I have the following definition of ...
1
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3answers
56 views

Estimating the sum of reciprocals of products of two primes

It's rather well-known that $$ \sum_{p \leq X} \frac{1}{p} \sim \log \log X,$$ where this is a sum over the positive integer primes. Can we efficiently estimate the sum $$ \sum_{p,q \leq X} ...
6
votes
2answers
82 views

Do the sum of all prime reciprocals with the digit $3$ converge or diverge?

$$\frac{1}{3}+\frac{1}{13}+\frac{1}{23}+\frac{1}{31}+\frac{1}{37}+\frac{1}{43}\cdots$$ Intuitively, I feel that this sum converges, but I really don't know why, (or if I am correct). Can I have a ...
3
votes
3answers
71 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
3
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1answer
34 views

In the comparison test, why are the summands in both the smaller and larger series required to be non-negative for all n?

Is the algorithm not well-defined if either one of the series has negative summands? Thanks,
1
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1answer
83 views

Find Limit Using Lebesgue Dominated Convergence

I'm trying to find the following limits using Dominated Convergence Theorem, but can't seem to find a dominating function. Any guidance would be greatly appreciated! $\lim\limits ...
4
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1answer
91 views

Is the $L_\infty$ norm of the $\mathcal{l}_2$ norm of this sequence of functions finite?

I am interested in proving or disproving the following claim and am stuck. We define a series of functions with the following properties. For each $i\in \mathbb{N}$ let $f_i\colon \mathbb{R}^+ \to ...
1
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1answer
23 views

Convergence in probability of a composite function.

Question: Let $f$ be a continuous function on $\mathbb{R}.$ If $X_n \to X$ in probability, then $f(X_n) \to f(X)$ in probability. The result is false if $f$ is merely Borel measurable. [Hint: ...
6
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1answer
92 views

Radius of convergence in power series $\sum_{n=0}^{\infty}(-1)^nx^{2^n}$

Given the series $$\sum_{n=0}^{\infty}(-1)^nx^{2^n}$$ determine the radius of convergence, and what can we say when $x=R$ and $-R$? Is it a power series? Power series should have the form of ...
0
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4answers
44 views

determine radius of convergence of one series

Given series $$\sum_{n=0}^{\infty}(-1)^nx^{n^2},$$ how can we determine the radius of convergence of this series? When $x=R$ or $-R$, what can we say?
0
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2answers
72 views

Does $\sum (-1)^k 2^{1/k}$ converge or diverge?

How am I supposed to determine the convergence of this series if I only know about the alternating series test and the divergence test?
5
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2answers
215 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
0
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0answers
21 views

Closure for Legendre Polynomials

Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates. For the derivation for such a problem, see these ...
0
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2answers
46 views

Rigor in Banach contraction principle

Given a contraction $T$ on $S$, we can after some triangle inequalities and so forth conclude that $\lim_{n \rightarrow \infty} \{T^n(x)\}$ converges to some point $x^*$ in $S$. I'm wondering: can't ...
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3answers
84 views

How to prove that this series converges? [closed]

Prove that $$ \sum_{n=2}^{\infty}\frac{\log (n+1)-\log n}{(\log n)^2}$$ converges.
3
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2answers
75 views

Proving convergence of a sequence $a_{n+1} = 3 - 2/a_n$ and finding the limit.

Let $(a_n)$ be the sequence defined by: $$a_1=\frac{3}{2}\qquad a_{n+1}=3-\frac{2}{a_n}\quad\text{for all }n.$$ Prove that the sequence is convergent. Calculate the limit of $(a_{n+1})$. ...
0
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0answers
28 views

Normal convergence of complex series

I have troubles with this task: Let $\mathbb{R}\_$ be the set of non-positive real numbers and $U = \mathbb{C}\backslash \mathbb{R}\_$ For $n \ge 0$, consider a function $f_n$$:U \rightarrow ...
2
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0answers
35 views

How does quadratic convergence imply linear convergence?

Linear convergence of a sequence is present if there exists a $c$ with $0<c<1$ such that $$|x_{k+1}-x| \leq c|x_{k}-x|, k=0,1,...$$ with $x$ being the limit of the sequence. Quadratic ...
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4answers
50 views

Convergence of a sequence $a_{n+1}=\frac{2(2a_n+1)}{a_n+3},n=1,2,…,a_1=1$

I have simply checked first five terms from where it is obvious that its limit is $L=2$, thus it is convergent sequence. I am interested in how to prove by induction that sequence is bounded and is ...
0
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1answer
29 views

Limit of ratio of sequences

Let $\{ a_n\}$, $\{ b_n\}$ be two sequences where $b_n$ is increasing such that $ \lim_{n \rightarrow \infty}b_n = \infty$. Also that $$\lim_{n \rightarrow \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n} = ...
1
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2answers
97 views

convergence of $\sum^\infty_{k=1} \frac {\ln(k)}{k^p}$

For what values of p does $\sum^\infty_{k=1} \frac {\ln(k)}{k^p}$ converge? Here is my work: $\ln (k) < k$ on $[1,\infty)$ so $\frac {\ln (k)}{k^p} < \frac {k}{k^p}$ Therefore, $ ...
4
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2answers
55 views

convergence of $ \sum_{k=1}^\infty \sin^2(\frac 1 k)$

convergence of $ \sum_{k=1}^\infty \sin^2(\frac 1 k)$ How can I do this? Should I use the Ratio Test (I tried this but it started getting complicated so I stopped)? Or the Comparison test(what ...
1
vote
3answers
101 views

How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?

How do I Find a closed form expression for : $$\sum_{k=0}^n \frac{(x-1)^k}{k+1}$$ Note :I have no idea how to do that, I am bad at evaluating series when we cannot use some standard series to do it. ...
2
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1answer
29 views

Sequence Involving Dirichlet Function

The question I have to prove is the following: Let $D(x)$ be Dirichlet Function: $$D(x) = \begin{cases}1 & x\in \Bbb Q \\ 0 & x \notin \Bbb Q \end{cases}$$ Let ...
1
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1answer
29 views

Criterion for series convergence - is there a rule such that?

I didnt find it on the internet, but I remember that I saw it somewhere. There are the known tests: if $$\lim _{n\to \infty }\sqrt[n]{a_n}\:<1$$ then $$\lim _{n\to \infty }a_n\:=\:0$$ if $$\lim ...
0
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1answer
20 views

Are these partial sums and partial products absolutely convergent?

For arbitrary $m \in \mathbb{N},$ $$\sum_{n=1}^{m}\ \sum_{d | \#_n}\mu(d)=\sum_{n=1}^{m}\big | \sum_{d | \#_n}\mu(d)\ \big |\ = \ 0,$$ $$\prod_{n=1}^{m}\ \prod_{d | \#_n}d^{\mu(d)}=\prod_{n=1}^{m}\big ...
1
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1answer
44 views

Why are higher order derivatives linked to the higher orders of convergence?

My textbooks defined the rider of convergence as follows (original image link) For an iterative process of the form $x_{n+1} = g(x_n)$, the order of convergence is first order when $|g'(x)| ...
2
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4answers
61 views

What is the correct radius of convergence for $\ln(1+x)$?

My text tells me this: And, Wolfram tells me this: Now, I'm not certain what to believe, but I believe I'm not certain because I'm not certain if Wolfram is using the logarithm with base $10$. ...
2
votes
2answers
56 views

The limit of a composed function

Let $g_n: \mathbb{N} \rightarrow \mathbb{R}$ and $f_n(x): \mathbb{N\times R} \rightarrow \mathbb{R}$. If $g_n \rightarrow g$ and $f_n(g) \rightarrow f(g)$, can we deduce $f_n(g_n) \rightarrow f(g)$? ...
3
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1answer
125 views

For which values does this series converge?

p and k are real numbers. For which values of p and k does the following double series converge $$\sum_{n,m=1}^\infty \frac{1}{n^p + m^k}$$ I am trying to find a better (and quicker) way to solve ...
0
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2answers
25 views

Looking for example of point wise convergent continuous bounded sequence of functions whose limit is neither continuous nor bounded

I am looking for a sequence of real valued functions $\{f_n(x)\}$ with domain some subset of $\mathbb R$ such that each $f_n$ is bounded , continuous and $f_n$ converges point-wise to some function ...
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0answers
28 views

How do I prove that $f(s)=\sum a_{n}{n}^{− s}$ converge for $Re(s)>0$if the partial sum of $a_{n}$ are bounded?? [duplicate]

let $f(s)$ be a power series defined as follow :$$f(s)=\sum a_{n}{n}^{− s}$$ Assume the partial sum of $a_{n}$ are bounded .My question here is : How do I prove that $f(s)$ converge for ...
0
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1answer
20 views

Radius of convergence of a power series (little question about power + constant)

My power series are: $$\sum _{n=1}^{\infty }\:\frac{x^{3n+1}}{\left(1+\frac{1}{n}\right)^{n^2}}$$ So its isnt difficult if it was written without the $+1$ in the power: $$\sum _{n=1}^{\infty ...
0
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0answers
36 views

help with investigating the uniformly convergence of a function sequence.

I have to check the uniform convergence of the below mentioned function sequence: $f_n(x) = \frac{1-\ln x}{nx}$ while $0<x<1$ at the answers, it's told that the sequence doesn't converge ...
5
votes
2answers
72 views

Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
3
votes
3answers
87 views

Determining the value to which the sequence $a_n=\frac{n!}{n^n}$ converges.

How can it be deduced that the sequence $a_n=\dfrac{n!}{n^n}$ converges to $0$? I can reasonably infer this to be true, because I see the pattern as $n$ approaches larger values, but I am unsure of ...
0
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1answer
39 views

Evaluating the convergence of the sequence $\{a_n\}=\frac{(-1)^{n-1}n}{n^2+2}$.

Set the sequence $a_n$ such that $\{a_n\}=\dfrac{(-1)^{n-1}n}{n^2+2}$. If $|a_n|$ converges (only to $0$, it would seem; correct me if I'm wrong), then $a_n$ must too converge, both to some value $L = ...