Convergence of sequences and different modes of convergence.

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Investigate convergence of a series [on hold]

$$\sum_{n=2}^{\infty} \left(2\cdot(-1)^{\frac{n^2(n-1)}{2}}-1\right)\cdot\frac{1}{2n-7\sqrt{n}}$$ I need to investigate convergence of this series, and I have no idea which criterion will be the most ...
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1answer
34 views

Convergence of a series and sum [on hold]

I have the followin series: $\sum_{n=2}^\infty\left(\sum_{k=2}^n(-2)^{-k}3^{-n+k}\frac{1}{k!}\right)$ I need to investigate convergence of the series and calculate its sum. How to do this (both)? In ...
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0answers
38 views

Generators: Analytic Vectors

Given a Banach space $E$. Consider semigroup $T:\mathbb{R}_+\to\mathcal{B}(E)$. Define its derivative: $$A_Tx:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)$$ (The domain being those elements whose limit ...
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1answer
43 views

Prove that if {$a_n$ } is a sequence that converges to A and $a_n$ ≥ 0 for all n then A ≥ 0.

Prove that if {$a_n$} is a sequence that converges to A and $a_n$ $\geq$ 0 for all n, then A $\geq$ 0. I have assumed on that contrary that A < 0. Pick $\epsilon$ = |A| > 0. Now, |$a_n$ - A| = ...
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2answers
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Which test to choose for these series and why?

Which convergence test we need to choose for these two equations? $$\sum_{k=1}^{\infty}\frac{k}{10+k^{2}}\tag{1}$$ $$\sum_{k=1}^{\infty}\frac{1\cdot3\cdot5\cdots(2k+1)}{4^{k}\, k!}\tag{2}$$ For ...
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1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
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2answers
40 views

Convergence of a certain series of Primes

This is a problem from Alan Baker's Comprehensive Course in Number Theory. We have to show that $\displaystyle \sum\limits_{p} \frac{1}{p (\log\log p)^{\delta}}$ converges for all $\delta >1$.Here ...
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546 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? 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 ...
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0answers
30 views

sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
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0answers
35 views

Uniform and absolute convergence of $\frac{1}{n^2+z^2}$

Let $z \in \mathbb{C}.$ I am asked to prove that $\sum\limits_{n=0}^{\infty} \frac{1}{n^2+z^2}$ converges on the set $\mathbb{C} \backslash \{ni : n\in \mathbb{Z}\} $. And also to prove that the ...
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1answer
31 views

Is this true?: converging sequence question

Let X be a metric space, and let A ⊂ X. Suppose that {pn} is a sequence in A which converges to some point p ∈ X. True or false: i) p ∈ A′ (limit points of A) (ii) p ∈ closure(A) These are both true, ...
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1answer
82 views

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$ implies $\lim_{n \to \infty} \int_B f_n \, d\mu = \int_B f \, d\mu$ for $B \subseteq X$

I'm having trouble with the following problem. Let $(X, \mathcal{M},\mu)$ be a measure space, where $X = [a,b] \subset \mathbb{R}$ is a closed and bounded interval and $\mu$ is the Lebesgue measure. ...
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1answer
52 views

Problem 3.14(e) in Baby Rudin

If $\{s_n\}$ be a sequence of complex numbers, define its arithmetic mean $\sigma_n$ by $$\sigma_n \colon= \frac{s_0 + s_1 \cdots + s_n}{n+1} \, \, (n = 0, 1, 2, \ldots). $$ Put $a_n = s_n - s_{n-1}$ ...
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1answer
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Convergence of sums of indicator random variables

I'm working through some practice problems for my final exam and I would like to get some ideas on tackling this problem: Let $(\Omega,\mathcal{F})=(\mathbb{R}_+,\mathcal{B}(\mathbb{R}_+))$ and ...
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1answer
13 views

Independent random variables convergence almost surely

I'm preparing for a final exam and I'm working through a practice exam. I'm somewhat stuck on a problem. The random variables $\{X_i\}$ are all independent and all satisfy $E[X_i^4]\leq 1.0$, but ...
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2answers
41 views

For what sequences $a_n$ does the sequence $(1+\alpha a_n)^n$ converge?

We know $ (1+\alpha/n)^n \rightarrow e^{\alpha} $ when $n\rightarrow +\infty$. Suppose we are given a modified version of the problem: $$ \quad (1+\alpha\cdot a_n)^n \tag{1} $$ The question ...
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1answer
53 views

Does the sequence $f\chi_{E_n^c}$ converge pointwise to $f$ if the measure of $E_n$ tends to zero?

If $(X,\mathcal{M},\mu)$ is a measure space, $f:X\rightarrow \mathbb{R}$ are such that $f\geq 0$ and $E_n\subseteq X$ measureable are such that $\mu(E_n)\rightarrow 0$, how would I be able to show ...
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3answers
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convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
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5answers
162 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
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1answer
33 views

If $f_n \to f$ in $L^p$, then prove that this sequence converges to $f$

Let $f_n \in L^p(\Omega)$ be a sequence that converges to $f$ in $L_p(\Omega)$. If $\Omega_n$ is a subset of $\Omega$ such that $\displaystyle{\lim_{n\rightarrow \infty}}\Omega_n=\Omega$, prove that ...
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Study the convergence of $\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$

Can you help me to study the convergence of the following series: $$\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$$ Thanks.
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Integrability: Cauchy Sequence

This thread is related to: Spectral Measure: Dominated Convergence Given a measure space $\Omega$. Consider a sequence of square integrables: $\int|f_n|^2\mathrm{d}\mu<\infty$ Suppose pointwise ...
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1answer
12 views

Infinite sequence of real numbers converging to x and y

So the question is: Suppose $x_i$ and $y_i$ are infinite sequences of real numbers converging to x and y. Show that $(x_i + y_i)$ converges to $x+y$. Show that $x_iy_i$ converges to $xy$. Here's ...
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0answers
28 views

Convergence of a complex function

I need to proof if the following function is bounded and convergent. $f(n)=\left(\frac{10+in}{n^{2}+2in}\right)^{n}$ Status: This should be correct. Can anybody confirm this? I tried it with ...
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1answer
29 views

Practice with closed sequences

Okay I have a final coming up, but I realized that I am still not adept at writing proofs. *Prove that if $\lim_{n\to \infty}p_n = p$ in a given metric space then the set of points {$p, p_1, p_2, ...
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Convergent sequence and index (N)

Okay so this problem looks simple, but I can't seem to get my notation right or something. The math tutor won't show me where i made the mistake and just goes on saying "that's not how you learn ...
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1answer
38 views

Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
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1answer
2k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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1answer
42 views

Convergence of a sequence in $l_2$

I am wanting to disprove (show that it is not the case) that for a sequence ${x_n} = x^n$, ($n\in\mathbb N$), that if $(x_i)^n \to x_i$ in $\mathbb R$,then $x^n\to x$ in $\ell_2$. I have gotten as far ...
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4answers
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Does $\sum_{n=1}^{\infty}\frac{\cos\left(\frac{n\pi}{2}\right)}{\sqrt{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^{\infty}\frac{\cos\left({\frac{n\pi}{2}}\right)}{\sqrt{n}}$$ The $\cos$ function: alternates between (-1) and 1 for every $n$ that is even. ...
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0answers
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Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
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1answer
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A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
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Sequence $\frac{x_{n}}{n} \sim Bin(1, \frac{1}{n^{2}})$. Show $x_{n}$ converges to $0. [closed]

Sequence $\frac{x_{n}}{n} \sim Bin(1, \frac{1}{n^{2}})$. Show $x_{n}$ converges to $0. Please help me with it, thanks very much
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If a function is convergent and periodic, then it is the constant function.

I have to prove that if a function f is convergent : $$ \lim\limits_{x\to +\infty} f(x) \in \mathbb{R}$$ and f is a periodic function : $$\exists T \in \mathbb{R}^{*+}, \forall x \in \mathbb{R}, f(x + ...
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1answer
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The link between the monotony of a function, and its limit

Let's assume I have a convergent function f, as x approaches to $$+\infty$$. Is-it true to say that it exists a real x0, such that forall x>x0, f is either increasing or constant or decreasing ? (And ...
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1answer
29 views

Convergence of a nice serie

For which value of $a>0$ and $b>0$ does $$\sum_{n\geq 0}\frac{a^n2^{\sqrt{n}}}{2^{\sqrt{n}}+b^n}$$ converge? Obviously it does not when $b<1$, but i don't have any answer otherwise.
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1answer
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79
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How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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4answers
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Convergent series with general term $a_nb_n$

I have the following question. Suppose that $\sum_{n=0}^\infty a_n$ is a convergent series, with $a_n > 0$, and suppose that $b_n > 0$ is a bounded sequence. Then show that the series ...
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1answer
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matlab program help

Wanting to write a matlab program to solve the following iteration: $x^{(k+1)}=b+\alpha\begin{bmatrix}2&1\\1&2\end{bmatrix}x^k,k=0,1,2,\cdots$ where alpha is a real constant. Find the values ...
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1answer
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Convergence of functions with different domain

Question: Is there a concept of convergence for functions $f_n: D_n \rightarrow X$ with different domains to a function $f: D \rightarrow X$? I know concepts like uniform convergence or almost ...
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Explore the convergence of improper integrals [closed]

I have troubles with tasks about improper integrals. 1)Explore the absolute and conditional convergence $\int_0^{+\infty} x^2 \sin(\frac{\cos(x^3)}{x+1}) {dx}$ 2)Find all $a$ for which the integral ...
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Cesàro summability and $\sum n \lvert a_n\rvert ^2 < \infty$ implies convergence

How can I prove that if $\sum_{n=1}^\infty a_n$ is Cesàro summable and if $\sum_{n=1}^\infty n |a_n|^2 < \infty$, then $\sum_{n=1}^\infty a_n$ converges?
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Determine the convergence of the following series

$$\sum_{k=0}^\infty {3^{k\ln k} \over {k^k}}$$ I need to determine the convergence of this series. I know it diverges, but how do I prove this?
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Find the interval of convergence of the power series. Be sure to include a check for convergence at the endpoints of the interval.

Find the interval of convergence of the power series. I know you use the power series and start with ratio test , test the points at the end.
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Which sequences are in the set defined by the convergence of $\sum_{i=1}^{\infty}|x_{i}|^{2}$?

$\operatorname{Map}(\mathbb{N}, \mathbb{R})$ is defined as the set of all mappings from $\mathbb{N}$ to $\mathbb{R}$ such that. Also, $$ V_{1} = \left\{(x_{i})_{i \in \mathbb{N}} \in ...
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1answer
25 views

Show almost everywhere convergence for variable with Chi distribution

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$R_n = \sqrt{X_1^2 + \ldots + X_n^2}$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I have tried to set ...
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1answer
59 views

How to construct a subsequence that is not the sequence with this specific critera and increasing indices? [on hold]

Say that $x_n \nrightarrow x$. I want to create a subsequence $x_{n_k}$ so that this subsequence has a subsequence $x_{n_{k_j}} \nrightarrow x$. (Reminder for non-convergence: If $x_n \nrightarrow x$, ...
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0answers
9 views

Almost sure convergence of Chi-Squared variable

Setting $$X_1,X_2,\ldots \overset{d}{\sim} \mathcal{N}(0,1)$$ $$S = X_1^2 + \ldots + X_n^2$$ I would like to show $\frac{R_n}{\sqrt{n}} \rightarrow 1$ almost everywhere. I am using Borel-Cantelli ...