Convergence of sequences and different modes of convergence.

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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} ...
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121 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 ...
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23 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 ...
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1answer
69 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 ...
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1answer
55 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 ...
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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 ...
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61 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} ...
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96 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 ...
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3answers
73 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: ...
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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,
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86 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 ...
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1answer
138 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 ...
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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: ...
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100 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 ...
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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?
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74 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?
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234 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 ...
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25 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 ...
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2answers
47 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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85 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.
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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 $(a_{n+1})$. ...
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30 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 ...
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39 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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67 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 ...
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30 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} = ...
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99 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, $ ...
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56 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 ...
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3answers
110 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. ...
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1answer
34 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 ...
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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 ...
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1answer
25 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 ...
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47 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)| ...
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68 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 $10$. ...
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58 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 $f_n(g_n) \rightarrow f(g)$? ...
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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 ...
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31 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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1answer
22 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 ...
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37 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 ...
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2answers
88 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 ...
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88 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 ...
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41 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 = ...
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80 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
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50 views

Determine whether $\sum \frac{1}{n^3 \ln(n^4+9)}$ converges

For the series $$\sum_{n=2}^{\infty}\dfrac{1}{n^3 \ln(n^4+9)},$$ I was thinking of using the limit comparison test with $1/n^3$?
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75 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{(3n+8)!}$

Determine whether or not this series converges or diverges $$\sum_{n=1}^{\infty}\dfrac{1}{(3n+8)!}$$ My attempt: I used the ratio test and ended up having ...
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1answer
50 views

Determine whether the series converges or diverges?

For the series $$\sum_{n=3}^{\infty}\dfrac{3n^2+8n}{7n^3-4n^2+11},$$ I was thinking of using the limit comparison test to $\dfrac{1}{n}$ but is there a better way?
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17 views

Can I show that a process which a supermartingale above a certain value and a submartingale below it converges?

In my work, I have many times encountered dynamic stochastic systems which are a submartingale (increasing in expectation) below a certain value of the variable, $X^*$ and a supermartingale ...
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20 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
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1answer
63 views

what is the radius of convergence of power series $\frac{z^2}{z}$?

I have a power series and am being asked to find its radius of convergence, but its structure of type $$\sum\frac{z^{2n}}{z^n}$$ is confusing me. How do I calculate radius of convergence of this power ...
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2answers
61 views

Prove convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$

Prove the convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$ This was a question on an exam. I needed to prove that the above integral converges using the comparison test. I thought about ...
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2answers
173 views

Help with Convergence of a series with sin and log

I tried to figured it out if the follwing series converges or not $$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\ln^2n}\ (-1)^{n}$$ I tried to show that $\sin(\frac{1}{n})$ is a monotonic but I'm ...