Convergence of sequences and different modes of convergence.

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2
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1answer
34 views

Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
0
votes
1answer
28 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
0
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2answers
27 views

Convergent series with condition

There exists a sequence of strictly decreasing real positive numbers $x_n$ such that its series converges, but the quantity $$\frac{x_n^2}{x_n-x_{n-1}}$$ doesn't converge to zero? All the famous ...
1
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2answers
46 views

Radius of convergence of $\sum_{n\geq 0}a_{n}x^{n}$.

Consider a series $\sum_{n\geq 0}a_{n}x^{n}$ where $a_{0}=2/3$ and $a_{n}=2-(1/2)a_{n-1}$ for all $n$. It is assumed that $2/3\leq a_{n}\leq 5/3$ for all $n\geq 1$. My problem is about determining its ...
0
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3answers
35 views

convergence of series $\sum_{n=1}^{\infty}\frac{4n^2+5n}{n(n^2+1)^{\frac{3}{2}}}$

I can't find a way to decide whether it convergent or not... tried the root test, but the result is 1(which means nothing)
-1
votes
2answers
40 views

for what valus of p, the series converge [closed]

For which values of $p>0$ does the series $\sum_{n=2}^\infty \frac{1}{n(\ln(n))^p}$ converge?
1
vote
1answer
17 views

Show that the radius of convergence of a sum of series is at least as big as minimum of radii of these series.

I am struggling with the following task. Suppose $\sum^{\infty}_{n=0}a_nx^n$ has radius of convergence $R$ and $\sum^{\infty}_{n=0}b_nx^n$ has radius of convergence $S$. I want to show that the ...
0
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0answers
11 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
0
votes
1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
1
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2answers
69 views
+50

Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} ...
2
votes
1answer
48 views

Prove that $\sum _{ n=1 }^\infty \left[ \frac { p(p+1)\cdots(p+n-1) }{ q(q+1)\cdots(q+n-1) } \right] ^\alpha $ converges

i need to prove this $$\sum _{n=1}^\infty \left[ \frac { p(p+1)\cdots (p+n-1) }{ q(q+1)\cdots (q+n-1) } \right]^\alpha, \qquad (p>0,q>0)$$ converges if and only if $\alpha (q-p)>1$ I ...
1
vote
1answer
28 views

Does the following sequence of random variables converge?

Let $X_1,X_2,...$ be independent random variables with $P[X_n=0]=1-1/n$, $P[X_n=1]=1/2n$, $P[X_n=-1]=1/2n$ Does $X_n$ converge almost surely? , Does $X_n$ converge in probability? I just started to ...
3
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1answer
51 views

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
0
votes
2answers
32 views

Series: only the latter terms matter

I've been told that when it comes to uniform convergence of series, only the tail matters, This seems intuitively obvious, but is there a theorem one can refer to? Further, if $\sum_{m}^\infty ...
13
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2answers
106 views

Suppose $\sum x_n$ converges (not necessarily absolutely), does $\sum \sin x_n$ necessarily converge?

The question is: If $\sum x_n$ converges, does $\sum \sin x_n$ converge? I know that if $\sum x_n$ converges absolutely, then $\sum \sin x_n$ converges. My intuition is that we cannot completely ...
0
votes
1answer
15 views

Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
0
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0answers
18 views

Expression about $\sum_{d=0}^{\infty} e^{dt}\frac{(5d)!}{(d!)^5}5\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right)$ [Done]

The purpose of my question in (here)[converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $ was actually to re-expression of following computation. ...
5
votes
1answer
55 views

converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $

\begin{align} \sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) = \sum_{n=0}^{\infty}\frac{4d}{(n+d+1)(n+5d+1)}= ? \end{align} I know from the $p$-test, ($i.e$ $\sum \frac{1}{n^p}$ ...
1
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0answers
23 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
-1
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0answers
47 views

Use the Lipschitz estimate to prove [closed]

Let ($f_n$) be a sequence of functions that are continuous on $[a, b]$ and differentiable on $(a, b)$. How to Use the Lipschitz estimate to prove that $|f_n(x) - f_p(x) - (f_n(c) - f_p(c))| \leq ...
0
votes
2answers
59 views

A series with the recursive formula.

A sequence $\lbrace a_{n}\rbrace_{n\geq 0}$ is constructed by choosing a value of $a_{n}$, and then the following elements are determined from the equation $a_{n}=2-\frac{1}{2}a_{n-1}$ for ...
1
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2answers
25 views

Bounding a summation by an integral

In progression of another question, my lecturer states $$\sum_{k=N+1}^\infty \frac{1}{k^2} \le \int^\infty_N \frac{1}{x^2}\, {\rm d}x.$$ However we have not covered bounding summations by integrals ...
0
votes
1answer
20 views

Does $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$ imply convergence of $\mathbf{x}^{(n)}$?

A sequence $\{\mathbf{x}^{(n)},n=1,2,...\}$. If $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$, does it also imply the convergence of the whole sequence $\{\mathbf{x}^{(n)}\}$?
0
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2answers
51 views

Prove the convergence of series $\sum_{k=1}^{\infty}\log(1+\frac{1}{\sqrt{k}})$ by Cauchy criterion

Given $$\sum_{k=1}^{\infty}\log\left(1+\frac{1}{\sqrt{k}}\right)$$ and by definition I need to prove that for $\forall \epsilon>0, \exists n_0 \text{ s.t. } \forall n>n_0, \forall p=1,2,...$ ...
2
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1answer
58 views

The series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$; the convergence and the differentiability.

Consider the series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$ for $x>-1$. a) Show that the series is pointwise convergent. Answer: I actually don't know how to show ...
0
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1answer
13 views

on the convergence of an infinite series involving logarithms

It looks like the following quantity $$ q(k)=\frac{k+1}{2k}(1+\log k) - \sum_{i=2}^k \frac{i}{k^2} \log i $$ tends to $3/4$ as $k$ goes to infinity. Is there a nice way to prove it?
2
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2answers
23 views

Does one of these conditions on a sequence imply the other one?

Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think ...
2
votes
1answer
25 views

A series of a function: comparison test?

Consider $\sum a_nx^n$. I wish to determine a convergence radius, but the $a_n$s do not behave nicely so root and ratio tests didn't get me anywhere. I can bound them from above and from below. Is ...
0
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2answers
54 views

Improper integral problem.

How to find divergence/convergence condition for $p$ on $$\int\limits_{2}^{\infty} \frac{1}{{(\ln x)}^p} \, \mathrm d x$$ I tried comparison test , but failed.
0
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0answers
18 views

Mode of convergence for partial Fourier series in $B( L_p[-\pi; \pi ])$, $p \in [1; \infty]$

Which mode of convergence takes place, strong, weak, or in norm? If we have sequence of continuous linear operators in $L_p[-\pi; \pi]$: $(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) ...
1
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1answer
23 views

Generalization of the Weierstrass M-test?

I had a co-student tell me that if we have $\sum_{n=1}^\infty v_n(x)$ and $\sum_{n=1}^\infty M_n$, and if the latter dominates the former for big enough $n$, then Weierstrass' M-test applies. But ...
0
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1answer
25 views

pointwise convergence of series?

We have just been introduced to infinite series of functions, and immediately started working on uniform convergence and Weierstrass' M-test, and how the sum of the series behaves in terms of ...
7
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3answers
266 views

Convergence of an alternating harmonic series

Consider the series $$\sum_{n=1}^\infty c_n \cdot \tfrac 1n$$ where $c_n$ is either $-1$ or $1$. In my case I have $$c_n = \begin{cases} 1&; \lceil np \rceil - \lceil (n-1)p \rceil = 1 \\ -1 ...
1
vote
1answer
22 views

Subset of the sequence space that's closed and bounded but not compact

Consider the sequences space $l^1 = \{a = (a_n)_{n \in \mathbb{N}_0} \subset \mathbb{C}, \sum_{n = 0}^\infty|a_n|< \infty\}$ with the norm $||a||_1 = \sum_{n = 0}^\infty|a_n|$. I want to show that ...
1
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1answer
16 views

Proving convergence for series containing ln and factorial

I am trying to show whether or not the following series ${a_n}$ converges. Based on the hint, I have tried using Bertrand's test, but I am having a hard time simplifying the absolute value of the ...
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1answer
27 views

Integral divergence

I´m trying to solve this problem about integral convergence, and I would be happy for any help. I shoul find out for what values of $a$ is this integral convergent: $$\int_0^\infty ...
3
votes
0answers
47 views

$\sum\limits_{n=1}^{\infty}\frac{\sin(nx)}{n^{\alpha}}$ does not converge uniformly for $\alpha \in (0,1]$

The question is, why $\sum\limits_{n=1}^{\infty}\frac{\sin(nx)}{n^{\alpha}}$ does not converge uniformly for $\alpha \in (0,1]$. I have a solution, but I dont understand the estimates, sorry=(. ...
0
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0answers
23 views

convergence, contracting sequence

I need a little help on the following question: let $x(0)$ be an element of reals and define a sequence $\{x(n)\}$ by: $x(1) =f(x(0)), x(2)=f(x(1)),...., x(n+1)= f(x(n))...$ show that if $m>0, ...
0
votes
1answer
50 views

Does the condition ${(n +1)^2} |x_n - x_{n+1}| \to 0$ imply that $\lim x_n$ exists?

Does the following condition imply the convergence of the sequence ${(x_n)}$ ? Given $ε > 0$ there exists an $n_0 \in \mathbf{N}$ such that $n > n_0$, ${(n +1)^2} |x_n - x_{n+1}| < ...
0
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1answer
15 views

Time to stable phase for the classic Susceptible-Infected-Susceptible epidemic model

The classic Susceptible-Infected-Susceptible epidemic model is the following: Each node is in one of the two states: Susceptible or Infected: Susceptible->Infected->Recovered. Let s and i ...
0
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1answer
32 views

Sequence who converges to $\sqrt{a}$ for every $a\geq0$

If we have: $$x_n=\left\lbrace \begin{matrix} b\in\mathbb{R}\setminus \{0\} & ,n=1 \\ \dfrac{a+x_{n-1}^2}{2x_{n-1}} & , n>1\end{matrix} \right.$$ Then, is easy to prove that $(x_n)\to ...
0
votes
1answer
29 views

Proof of Partial Sum - Series

By considering the partial sums for $S$, that is $$S_n =1+2+3+···+n$$ show that the infinite series $S$ does not converge. How do we show that this does not converge? How to rigorously prove it ...
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0answers
27 views

Series Proof Question [duplicate]

By considering the partial sums for S, that is Sn =1+2+3+···n show that the infinite series S does not converge. However in this video http://www.numberphile.com/videos/analytical_continuation1.html ...
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2answers
65 views

Do the following series converge if $a_n>0$ and $ \sum_{n=1}^{\infty}a_n$ diverges?

Do the following series converge if $a_n>0$ and $\sum_{n=1}^{\infty}a_n$ diverges ? a.) $\sum_{n=1}^{\infty}{a_n \over 1+ a_n}$ b.)$\sum_{n=1}^{\infty}{a_n\over 1+ a_n ^2}$ ...
1
vote
4answers
42 views

If $x_n \to a$ and $x'_n \to a$, then $\{x_1, x'_1, x_2, x'_2, …\} \to a$

I know that if all subsequences of $\{x_1, x'_1, x_2, x'_2, ...\}$ converge to $a$, then $\{x_1, x'_1, x_2, x'_2, ...\}$ converges to $a$, but I only know two subsequences of $\{x_1, x'_1, x_2, x'_2, ...
1
vote
1answer
25 views

Uniform continuity preserves uniqueness of convergent sequences?

If $f: (a, b) \to \Bbb R$ is uniformly continuous, $\{x_n\}$ and $\{x'_n\}$ are sequences in $(a, b)$ with $x_n \to b$, $x'_n \to b$, $f(x_n) \to y$, and $f(x'_n) \to \overline{y}$, prove $y = y'$. ...
2
votes
1answer
39 views

How to show $ \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}})$ converges?

Let $a_{n} \ge 0 \hspace{1cm} \forall n \in$ $ \mathbb{N} \cup \{0\}$. and $ \sum_{n=1}^{\infty} a_{n}$ converges and $ b_{n}=\sum_{k=n}^{\infty} a_{k} $ Then we have to prove that$ ...
5
votes
1answer
35 views

Convergence of a sequence 4

Suppose there is a sequence $\{ x_n \}$. Let us define another sequence $\{ y_n \} $ such that $$y_n=2x_{n+1} - x_n$$ Please prove that if $\{y_n\}$ converges to $L$, then $\{x_n\}$ is convergent and ...
2
votes
1answer
56 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
2
votes
2answers
103 views

Integral test for convergence proof

Can someone help me understand this proof? I don't understand why $f(n+1) = \int_n^{n+1}{f(n+1)}$ Thank you so much and I am sorry I have nothing else to contribute as I'm fearing it is a ...