Convergence of sequences and different modes of convergence.

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Does convergence in distribution implies convergence of expectation?

If we have a sequence of random variables $X_1,X_2,\ldots,X_n$ converges in distribution to $X$, i.e. $X_n \rightarrow_d X$, then is $$ \lim_{n \to \infty} E(X_n) = E(X) $$ correct? I know that ...
7
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3answers
280 views

If $\frac{\sin{a_n}}{a_n}\rightarrow 1$ then $a_n\rightarrow0 $

I'm trying to prove that if $\;\frac{\sin{a_n}}{a_n}\rightarrow 1$ then $a_n\rightarrow0$, assuming $a_n\neq 0$ for all $n$. I think this is easy enough to show as follows: first, prove ...
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Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
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120 views

Convergence of $\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$

Does the series $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^{1+1/n}}$$ converge absolutely, converge conditionally, or diverge? I've tried applying the ratio test and the root test, and in both cases the ...
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3answers
240 views

If a fixed point is a limit of a subsequence of iterates, must the whole sequence converge to it?

Say $X$ is a compact metric space, with $f:X \to X$ continuous. Now, for any $x_0$, $f^n(x_0)$ must have a convergent subsequence, say $f^{n_i}(x_0) \to x_\infty$. If we know that any such limit is a ...
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1answer
186 views

Convergence in the Box Topology.

Given the sequence in $\mathbb{R}^\omega$: $$y_1=(1,0,0,0\ldots), y_2=(\frac{1}{2},\frac{1}{2},0,0\ldots), y_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\ldots),\ldots$$ I know that it converges in the ...
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237 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
7
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1answer
156 views

methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$

As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
7
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1answer
206 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
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1answer
515 views

Convergence of a product series with one divergent factor

I'm currently struggling with the following problem: Let $\displaystyle \sum_{k=1}^{\infty} a_k$ be a convergent series with $a_k \in \mathbb{R} \setminus \{0\}$. Then is there always a sequence ...
7
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1answer
309 views

Does the integral test work on higher dimensions?

The integral test of convergence states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty ...
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2answers
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Kummer's test - Calculus, Apostol, 10.16 #15.

I want to prove the following: Let $\{ a_n \}$ and $\{ b_n \}$ be two sequences with $a_n>0$ and $b_n>0$ for all $n \geq N$, and let $$c_n = b_n - \frac{a_{n+1}b_{n+1}}{a_n}$$ Then If there ...
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359 views

Disproving uniform convergence

Given a sequence of functions $f_n$ which is known to be pointwise convergent how would you go about showing that is not uniform convergent? The particular example I'm working with is $f_n ...
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3answers
212 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
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How will the limit $\lim \sin(A^{n})$ behave for $|A| > 1$?

While answering to the following question Problem. Find the range of $x$ for which $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \sin(x^{-n})$ converges. Unlike some careless answerers who merely ...
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122 views

convergence of series with $k!$

check if the following series converges: $\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$ where $k!!=k(k-2)(k-4)(k-6)...$ I came across this exercise while going trough some old exams. I'm ...
7
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1answer
180 views

What functions satisfy $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ^{\infty} f(x_n) < \infty $ [duplicate]

Could you help me with this problem? What functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following implication? $\sum _{n=1} ^{\infty} x_n < \infty \ \Rightarrow \sum _{n=1} ...
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1answer
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Question dealing with a series of functions that uniformly converges on $[0,1]$

Let $f_1, f_2, \ldots$ be a sequence of continuous positive functions of $[0,1]$ and let $a_n = \sup\{ f_n(x) : x \in [0,1]\}$. In class, we showed that if $\sum f_n$ uniformly converges on $[0,1]$, ...
7
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1answer
122 views

$\Gamma(1/2-n+it)$ converges uniformly

Prove that $\Gamma(1/2-n+it)\rightarrow 0$ uniformly as $n\rightarrow\infty$ for $t\in\mathbb{R}$, where $n$ is a positive integer. I'm not sure which definition of $\Gamma$ would be easiest to ...
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Convergence of $\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$

If $\sum_{n=2}^{\infty}a_n$ converge, then also converge this series? $$\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$$ Please verify my answer below Counterexample: ...
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1answer
383 views

Uniform convergence of infinite series

Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
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1answer
134 views

Zeros in the complex plane and convergence

I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty. A text I am reading states the following: ...and given that ...
7
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2answers
106 views

Series convergence/divergence

I was trying to prove the following question. Part a is intuitive but couldn't give a clear mathematical argument. For parts b and c It seems there is something I am not seeing. Any help ? If ...
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4answers
350 views

What steps should I be doing to determine if this series is convergent or divergent?

The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$ The first thing I did was use the divergence test which didn't help since the result of the limit was 0. If I multiply it through, the result ...
6
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4answers
443 views

What is the sense of the equality sign =?

Sometimes in real analysis, we write an equality and specify that is true in the sense of the $L^2$ norm. My question is, when in mathematics we write '=' and don't specify anything, what we really ...
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4answers
353 views

Power Series with the coefficients $n!/(n^n)$

I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series. Or, by Cauchy Hadamard, the limit of $(n!/(n^n))^{(1/n)}$ as n approaches infinity. ...
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6answers
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Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
6
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3answers
314 views

Disprove uniform convergence of $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ in $[0,\infty)$

How would I show that $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ does not uniformly converge in $[0,\infty)$? I don't know how to approach this problem. Thank you.
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399 views

Convergence of Series

At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to ...
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7answers
169 views

Why $ \int^{+\infty}_{-\infty} x \, dx \neq 0 $

We are going over improper integrals and tests for convergence in my Calc II course. During a lecture, my professor warned us to take caution when taking integrals from negative infinity to infinity. ...
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4answers
116 views

Prove the limit for a series of products.

Let $\beta > 0$, $\lambda > 1$. Show the identity $$\sum_{n=0}^\infty\prod_{k=0}^{n} \frac{k+\beta}{\lambda + k + \beta} = \frac{\beta}{\lambda - 1}$$ I have checked the statement numerically. ...
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4answers
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proving convergence for a sequence defined recursively

The sequence $\left \{ a_{n} \right \}$ is defined by the following recurrence relation: $$a_{0}=1$$ and $$a_{n}=1+\frac{1}{1+a_{n-1}}$$ for all $n\geq 1$ Part 1)- Prove that $a_{n}\geq 1$ for all ...
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501 views

Convergence in metric and a.e

How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?
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Limit proof of a sqrt-heavy expression with binomial formula / sandwich-rule

Let be $$b_n := \sqrt{n+\sqrt{2n}}-\sqrt{n-\sqrt{2n}}, n\in\mathbb{N}$$ a sequence. I am to determine $\lim\limits_{n\to\infty}b_n$ which is obviously $\sqrt{2}$. My first step was to transform the ...
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Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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4answers
640 views

Terms that get closer and closer together [duplicate]

Possible Duplicate: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy? I was thinking about sequences where it appears the terms get closer and ...
6
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2answers
268 views

Prove that the sequence given by $c_n = \sqrt{1+c_{n-1}}$ converges and find the limit

Let $c_1 = 2$, and for $n > 1$, let $c_n = \sqrt{1+c_{n-1}}$. Prove: (by induction) that $c_n < 2$, for $n > 1$. (by induction) that {$c_n$} is monotonically decreasing. that ...
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2answers
326 views

Convergence of $\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$

A student was recently asked this question by his instructor: $$\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$$ Converge or diverge? I feel a little dumb for not being able to answer it. The ...
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Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$

I'd like your help with the following claim to prove: $$\lim_{n \to \infty} \int_{0}^{\sqrt n}\left(1-\frac{x^2}{n}\right)^ndx=\int_{0}^{\infty} e^{-x^2}dx.$$ I think I should use the claim: Let ...
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4answers
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Is this implication true? [duplicate]

Suppose that a real sequence $u_n$ is such that $$u_{n+1}-u_n \rightarrow0$$ That is not enough to prove that $u_n$ is convergent (take $u_n=ln(n)$) Now what if $u_n$ is bounded ? I guess it does ...
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3answers
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proving convergence of a sequence and then finding its limit

For every $n$ in $\mathbb{N}$, let: $$a_{n}=n\sum_{k=n}^{\infty }\frac{1}{k^{2}}$$ Show that the sequence $\left \{ a_{n} \right \}$ is convergent and then calculate its limit. To prove it is ...
6
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3answers
226 views

Sequence Convergence of $\sum_n\frac{(-1)^{n+1}}{3n + n(-1)^n}$

I have the following series $\displaystyle \sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{3n + n(-1)^n}$. Does it converge? I wanted to the alternating series test, but that's not easy because of the two ...
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252 views

Application of Central Limit Theorem

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
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204 views

Why does this series converge?

My question is: Why does the series $$ \sum_{j,k=1}^\infty \frac{1}{j^4+k^4} $$ converge? I tested the convergence with Mathematica and Octave, but I can't find an analytical proof. In fact, ...
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Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.
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$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Please help me check, if $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$ $f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
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341 views

Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$

Please help me to prove that $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$ converge and it's limit $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: ...
6
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3answers
91 views

Series comparisons and logarithms?

Prove the convergence of $$ \sum_{n = 1}^{\infty} {\sqrt{\, 2n - 1\,}\,\ln\left(4n + 1\right) \over n\left(n+1\right)} $$ I've been struggling for hours on this. By the textbook we have the limit ...
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3answers
272 views

Probability Puzzle : Robot and coins

Someone walks into your room and dumps a huge bag of quarters all over the floor. They spread them out so no quarters are on top of any other quarters. a robot then comes into the room and is ...
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2answers
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Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...