Convergence of sequences and different modes of convergence.

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Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
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31 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...
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0answers
18 views

Independence and limits [duplicate]

Suppose $X_n$, $Y_n$ are independent real valued random variables for every natural n. And that the limit as n tends to infinity almost surely exists finitely, say X, Y respectively. Is it necessary ...
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0answers
14 views

The inverse of a Dirichlet serie

When we consider a Dirichlet serie : $$F(s)=\sum_{n=1}^{+\infty}{\frac{f(n)}{n^s}}$$ which converges absolutly in the half-plane $\{ \Re(s) > \sigma \}$ with $f(1) \neq 0$, what can we say about ...
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2answers
24 views

Convergence of sum using D'Alembert.

I have to find the convergence of this series: $$\sum \limits_{n=0}^{\infty} \frac{(1+{\frac 1n})^n}{2^n}$$ I started by using D'Alembert: $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n}$, So : ...
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1answer
16 views

Showing the convergence of improper integral.

Hello I have to show that this improper integral is convergent: $$\int_{0}^{1} \frac{e^{\frac{-1}{x}}}{x^2} dx$$ , but I don't have any starting ideea. Any tips would be great, thank you.
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3answers
46 views

$\lim_{n \rightarrow \infty } {n \choose k} a^n = 0$

Let $k$ a fixed positive integer and $0<a<1$ a real number. Prove that $\lim_{n \rightarrow \infty } \left( \frac{n!}{(n-k)!k!}\right) a^n = 0$ . I stuck in this limit i remember that ${n ...
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2answers
51 views

Real Analysis Question on convergence

The question is: Show that if the partial sums $s_n$ of the series $\sum\limits_{k=1}^\infty a_k$ satisfy $\vert{s_n}\vert\leq Mn^r$ for some $r<1$, then the series $\sum\limits_{n=1}^\infty ...
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2answers
27 views

Rearrangement of absolutely convergent series is absolutely convergent??

If $\sum a_n$ is absolutely convergent, is it true that every rearrangement of $\sum a_n$ is also "absolutely" convergent? I know that by the Rearrangement Thm., every rearrangement of $\sum a_n$ is ...
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1answer
24 views

Holomorphic function and series converges in the unit disk $ (|z_{k}| < 1) $

$ f $ is holomorphic in the unit disc , bounded and not identically zero and $z_{1},z_{2},\ldots,z_{n},\ldots $ are its zeros$ (|z_{k}| < 1) $ , $a$ is a real number My question is :for which ...
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0answers
46 views

Sum of a power series

I have to find the sum of this series $$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$ Using integral, I got $$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$ I ...
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4answers
96 views

Power series convergence radius

My question is: how do I calculate the radius convergence of a power series when the series is not written like $$\sum a_{n}x^{n}?$$ I have this series: $$\sum\frac{x^{2n+1}}{(-3)^{n}}$$ Can I use the ...
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1answer
31 views

Question about limit of a product of two real sequences

Let $(a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\subseteq\mathbb R$. Moreover assume that $\lim_{n\to \infty} a_n=c_1\in\mathbb R$ with $c_1\neq 0$ and that $\lim_{n\to \infty} a_nb_n=c$ for some ...
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1answer
27 views

Convergence of random harmonic series

The problem is to show that the random harmonic series $X_n:=\sum_{n=1}^{\infty}\frac{\nu_n}{n}$ with $P[\nu_n = 1] = P[\nu_n = -1] = \frac{1}{2}$ converges. It is obvious that the harmonic series ...
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4answers
69 views

Prove $u_{n}$ is decreasing [closed]

$$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$ Prove it is decreasing and convergent and calculate its limit. Is it possible to define $u_{n}$ in terms of $n$? In order to prove it is decreasing, I ...
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1answer
31 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
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1answer
33 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
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1answer
34 views

Show that if a series of functions converges uniformly then it also converges in $L^2$ and pointwise sense. [closed]

Consider any series of functions on any finite interval. Show that if the series converges uniformly, then it also converges in the $L^2$ sense and pointwise. Need help with this problem thank you.
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2answers
83 views

convergence of a generalized Riemann integral

Could you please provide me some hints to test the convergence of this integral below? $$\int\limits_{0}^{+\infty}\dfrac{\sin x\cdot \sin 2x}{x^\alpha} \, dx$$ where $\alpha \in \mathbb{R}$
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26 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
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1answer
55 views

Convergence of tricky series

I've stumbled upon a particularly unpleasant series, and I can't quite seem to crack it. $\sum\limits_{n=1}^{\infty}\dfrac{\ln(1+nx)}{n^{2}} $ I need to show uniform convergence on any interval of ...
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1answer
26 views

A basic question about the radius of convergence of infinite power series.

I have a somewhat theoretical question to the definition of the radius radius of convergence of infinite power series. According to the definition for a power series $\sum_{n=0}^\infty a_nx^n$ radius ...
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0answers
60 views

Is this a geometric series?

A geometric series is, in general, defined by: $$ \sum_{k=0}^{n-1}a\cdot r^{k}=a\cdot\dfrac{1-r^n}{1-r},\quad\quad \quad\quad \quad\quad r\neq1 $$ If I have instead the following: $$ ...
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1answer
44 views

Convergence of $u_n$ defined by $\sum\limits_{k=1}^{u_n-1}\frac{1}k\leq n<\sum\limits_{k=1}^{u_n}\frac{1}k$

For all $n\in\mathbb{N}^*$, let $u_n\in\mathbb{N}^*$ such that $$\sum\limits_{k=1}^{u_n-1}\frac{1}k\leq n<\sum\limits_{k=1}^{u_n}\frac{1}k$$ Does the sequence $\dfrac{1}{u_n}$ converge (and how ...
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2answers
39 views

An explicit example of a sequence with two special convergent subsequences

I'm looking for an explicit example of a sequence $(u_n)_{n\ge 0}$ with values in $\mathbb{C}$ such that $(u_{2n})_{n\ge 0}$ and $(u_{3n+1})_{n\ge 0}$ are convergent but $(u_n)$ isn't. It is easy to ...
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2answers
71 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
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99 views

Does the series $\sum_{n=2}^{\infty} \frac{\cos(\ln(\ln(n)))}{\ln(n)}$ converge?

Could someone please give a hint about how to solve the convergence of the following series: $$\sum_{n=2}^{\infty} \frac{\cos(\ln(\ln(n)))}{\ln(n)}$$
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2answers
130 views

for which value of $a$ that $ \big(\sum\frac {1}{u_n}\big) $ converges?

For any real number $a$ (positive or negative), define a sequence $\{u_n\}$ (depending on $a$) recursively by $u_0=2$ and $$ \int_{u_n}^{u_{n+1}}(\ln u)^a\,du=1.$$ For which $a\in\mathbb{R}$ does $$ ...
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1answer
20 views

Determining the domain of convergence of a function

For $$f_n(x)=n\cdot \sin\left(\frac{x}{n}\right)$$ How do I determine for what values of x this series converge? (like $[a,b]$ or $(a,b]$...) thank you very much for any help!
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31 views

A simple convergent integral but not absolutely convergent.

Anybody knows a simple example for convergent function but not absolutely convergent? ( simple = easy ) Thanks for coments!!!
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1answer
18 views

Convergence in distribution problem

I want to prove that, in $(\mathbb{R},B(\mathbb{R}))$, we have that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\frac{i}{n}}$ converges to $U_{[0,1]}$. We need to prove, by definition, that $\lim_{n \to ...
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1answer
16 views

Convergence in $L^1$ of $X_n(z):=n^{\alpha}1_{[\frac{1}{n+1}, \frac{1}{n}]}(z)$

Let $X_n(z):=n^{\alpha}1_{[\frac{1}{n+1}, \frac{1}{n}]}(z)$ with $\alpha>0$ on $[0,1]$ be a series. I have two questions about this. For which $\alpha$ does $X_n$ converge in $L^1$ to 0 with $n ...
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2answers
87 views

Prove that $\{a_n\}_{n=1}^{\infty}$ converges to $\frac{x}{2}$.

Let $x$ be any positive real number, and define a sequence $\{a_n\}_{n=1}^{\infty}$ by $$ a_n=\frac{[x]+[2x]+\cdots+[nx]}{n^2} $$ where $[x]$ is the largest integer less than or equal to $x$. ...
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1answer
81 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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1answer
21 views

Subsequence to infinity proof

How to prove this: Every not bounded above sequence has subsequence which limit is infinity when n->infinity It's nearly the same what Bolzano–Weierstrass ...
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1answer
62 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
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1answer
24 views

Show convergence of sequence

Consider a sequence $(x_n)$ such that $\forall n \in \mathbb N, |x_{n+1}-x_{n}|\leq 2^{-n}$ How to show that it converges? Any hints would be appreciated
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If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
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1answer
14 views

Net-Complete $\iff$ Sequence-Complete

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
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1answer
27 views

Absolute or conditional convergence of alternated series.

I need to study the convergence (absolute or conditional) of this alternated series : $\sum\limits_{n=1}^{\infty}(-1)^n \left(\frac{n}{7n^2+3}\right)$ Here is what I did so far : $\lim\limits_{n ...
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1answer
42 views

Show that this difficult sequence converge

$x_1=\sqrt{2},\,x_2=\sqrt{2+\sqrt{2}},\, x_3=\sqrt{2+\sqrt{2+\sqrt{2}}},\, \cdots$ How to show that the sequence converges? We can say that it is monotonic but if it is bounded?
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1answer
22 views

Convergence in Probability and in Quadratic Mean for a sequence of random variables

I have been trying to determine whether a sequence of random variables, $X_1,X_2,\ldots,X_n$, such that $$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ...
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2answers
38 views

Arc length integration

Find the length of the arc formed by $x^2=10y^3$ from point A to point B, where A=(0,0) and B=(100,10). My attempt: $\int_0^{100} \! \sqrt{1+(\frac{2}{3x})^2} \, \mathrm{d}x. $ However this integral ...
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1answer
41 views

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if a) |$z$| $\le$ 2. b) |$z$| $<$ 2. c) |$z$| $\le$ $\sqrt{2}$. d) |$z$| $<$ $\sqrt{2}$. Please anyone give me the answer. I think ...
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2answers
272 views

Convergence of $\sum _{k=1}^\infty \sin \left(\sqrt{k}\right)/k$

This was asked at an oral examination. Does the series $\displaystyle \sum _{k\geq1}\frac{\sin\left(\sqrt{k}\right)}{k}$ converge ? After playing with Mathematica, it's very likely it ...
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4answers
48 views

Does this sequence converge

Does the sequence $x_n=3^n−2^n$ converge? I can show that it is increasing but how to show that it is bounded?
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1answer
34 views

If $f_n$ converges uniformly then $\cos(t)f_n$ converges uniformly?

In my Fourier Series course, it seems the following result is used: If a series of function $\sum a_n(t)$ converges uniformly then the sequence of functions $\cos(t)\sum a_n(t)$ converges ...
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1answer
36 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
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1answer
33 views

Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...
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2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...