Convergence of sequences and different modes of convergence.

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2
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1answer
65 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
0
votes
1answer
43 views

Asymptoting to 0: is erfc(z) quicker than exp(-z)?

If I have a function of the form $\mathrm{erfc}\left(z\right)/e^{-z}$, should I expect its limit at large $z$ to be $0$ or $\infty$? My instinct is that it should be $0$, by considering the ...
-3
votes
1answer
21 views

Convgergence of series with different parametes [closed]

I have this series : $$\sum\limits_{n=1}^{\infty} n^{-a}\log(n)^{-b}$$ For what values of $a$ and $b$ does the series converge?
1
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2answers
60 views

Does this series converge or diverge and by which test?

$$\sum_{n=1}^\infty (-1)^{n+1} \sin(1/n^3)$$ I tried to apply the divergence test. I know $\lim_{n\to \infty}$ is 0 for $b_n$ but I don't think $b_n$ is decreasing. any ideas on how I can test this ...
0
votes
1answer
15 views

Separability of $l^{p}$ spaces

How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$: $||x||_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$ Is a separable space? (i.e. showing that there is a ...
0
votes
2answers
69 views

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}\sin{(a_n)}$ converges.

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\displaystyle \sum_{n=1}^{\infty}a_n$ converges then $\displaystyle \sum_{n=1}^{\infty}\sin{(a_n)}$ converges. I conjecture that the final term ...
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 $\sum_{n=1}^{\infty}x_na_n$ ...
-1
votes
2answers
53 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?
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: ...
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 ...
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. ...
4
votes
3answers
561 views

Why is the convergence absolute?

There is one thing my book uses in a proof after Abels theorem which I do not understand: Lets say that $\sum_{n=0}^\infty a_n$ converges. For $0\le x<1$, we look at $\sum_{n=0}^\infty a_n x^n$. ...
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
336 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
3answers
100 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. ...
1
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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} ...
5
votes
2answers
211 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 ...
2
votes
3answers
81 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{∞} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{∞} \frac{\ln n}{n^2}$ to ...
0
votes
1answer
15 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 ...
0
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2answers
52 views

If I'm asked to prove $\{1/n\}_{n=1}^\infty$ converges to $0$, can I assume the euclidean metric?

Using the definition of limit if I'm asked to prove $\{1/n\}_{n=1}^\infty$ converges to $0$, can I assume the euclidean metric?
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 ...
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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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 ...
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: ...
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 ...
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} ...
1
vote
1answer
356 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
4
votes
1answer
88 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 ...
0
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1answer
360 views

Convergence of a sequence in trigonometric functions

Is the sequence $(a_n)$ defined as $$a_n :=\dfrac {\sin\Big( \dfrac {\pi}4+\dfrac n2 \Big)\sin\Big(\dfrac{n+1}2 \Big)}{\sin\Big(2n+2\Big)\sin\Big( \dfrac {\pi}4+\dfrac {n-1}2 \Big)\sin\Big(\dfrac{n}2 ...
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 ...
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 ...
3
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1answer
34 views
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 $(a_{n+1})$. ...
1
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1answer
73 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis I, specifically the sequence and series part. I've just come across the definition of convergence: A sequence $(a_n)$ converges to a real number $a$ if, ...
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: ...
0
votes
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?
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 $f_n(g_n) \rightarrow f(g)$? ...
1
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2answers
75 views

Does it converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get $1$. I suppose I should compare it with some other series, but I can't figure out ...
0
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2answers
71 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?
1
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1answer
613 views

Convergence: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Suppose one has: $$\|\varphi\|=\lim_n\|\varphi_n\|$$ Then it follows: $$\varphi\rightharpoonup\varphi\implies\varphi_n\to\varphi$$ How can I check this?
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 ...
6
votes
3answers
413 views

Nets and compactness in topological spaces.

I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove ...
0
votes
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.
1
vote
1answer
215 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that $lim(Arctan nx) = (pi/2)sgn x$ for $x$ in R, $x>=0$. I have a final coming up and I've started doing some ...
4
votes
1answer
593 views

Nested radicals

Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges. A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt ...
5
votes
3answers
108 views

An improper integral with parameter $\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$

I have to analyse the convergence of this integral: $$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in \mathbb{R}$. I have thought to write: ...
1
vote
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, $ ...
0
votes
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 ...