Convergence of sequences and different modes of convergence.

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49 views

Show that $\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ [duplicate]

$\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ Tried: here suppose N is poission distribution with parameter n $\lim_{n\rightarrow \infty} \sum_{i=0}^{n} ...
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2answers
33 views

What's wrong with my radius of convergence test?

Given $\sum \limits _{n=2} ^\infty \frac{(\ln x)^n} n$, find its radius of convergence $R$. Using the ratio test, I arrived at $$|\ln x| < 1 \implies e^{|\ln x|} < e^x \implies |x| < e ...
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1answer
14 views

Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be ...
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2answers
32 views

How can i chech the convergence of $ \sum_{i=0}^{ \infty} \frac{ \theta^n}{n^r} $ , $r >1$ and $ \theta \in (0,1) $?

How can i chech if the serie of $ \sum_{n=1}^{ \infty} \frac{ \theta^n}{n^r} $ , $r >1$ and $ \theta \in (0,1) $ is converge?
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2answers
40 views

For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
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1answer
27 views

Uniform convergence of a function composition

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
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35 views

What is the convergence value of series $\sum_{i=1}^{\infty} i^2 * (0.4)^i$

One technique to cope with some series is using derivation of a geometry series. But in this case I think $i^2$ makes this technique useless. Any idea would be appreciated.
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1answer
20 views

Is the integral test for convergence still applicable?

$$\sum _{n=0}^{\infty \:}\left(n\ e^{-n^2}\right)$$ Can I still use the integral test to determine whether this series converges or diverges given that $f(x) = x\ e^{-x^2}$ is not decreasing on the ...
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1answer
43 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
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1answer
26 views

Absolute convergence of ordinary Dirichlet series

I am currently reading Serre's 'A course in Arithmetic' and I have a question about proposition 8 of section 2.4 (but I think the question can be answered without knowing the book.) The proposition ...
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3answers
78 views

Check convergence of $\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$

Zoomed version: $$\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$$ So, I've seen similar example at Convergence or divergence of $\sum \frac{3^n + n^2}{2^n + n^3}$ And I liked that answer : ...
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0answers
24 views

A complex sequence of functions $(f_n)$ is continuously convergent iff it's compactly convergent against a continuous function

Let $G \subseteq \mathbb{C}$ be a region in $\mathbb{C}$, i.e. $G$ is open, nonempty and connected, and let $f_n: G \to \mathbb{C}$ be a sequence of complex-valued functions, with $n \in \mathbb{N}$. ...
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1answer
60 views

What sum to $\sum_{n=0}^\infty\frac{x^n}{(n+1)!}$ in its convergence radus?

My task is this: Find the sum to $$\sum_{n=0}^\infty\frac{x^n}{(n+1)!}.$$ in its convergence radus. My work so far: By ration test we get ...
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1answer
22 views

Convergence of $\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$

For which $\alpha$ and $\beta$ is the sum $$\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$$ $$µ\left(k\right) = \left\{ \begin{array}{lr} \alpha & : k\ is\ even\\ \beta & : k\ is\ ...
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23 views

Monotone Convergence Theorem in Measure Theory.

My textbook defined M.C.T. by for $\{f_k\}$ be a sequence of measurable functions on $E\subset\mathbb{R}^n$, If $f_k\nearrow{}f~~a.e.$ on $E$ and there exists $\phi\in{}L(E)$ such that ...
3
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1answer
44 views

Find the interval of convergence to $\sum_{n=2}^\infty\frac{(-1)^nx^{n}}{n(n-1)}.$

My task is to find the interval of convergence to:$$\sum_{n=2}^\infty\frac{(-1)^nx^n}{n(n-1)}.$$ My work so far: Taking $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=|x|<1\implies ...
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1answer
28 views

Find if the series $\displaystyle\sum_{n=1}^\infty\frac{\cos n\sin\frac{1}{n}}{n}$ converges

As far as I understand this is a Leibniz series therefore it's converging. What I was thinking is finding a way to "change" the $\cos n$ to $\cos {\pi n}=(-1)^n$ and get a ...
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40 views

asymptotic behavior of functions which are defined recursively

For $x\in[0,1]$, define $f_1(x)=1$ and $$ f_{n+1}(x)=(1-x)^n\left(nx+1\right)+f_n(x)\left(1-(1-x)^{n+1}\right)\;\;\mbox{for}\;\;n=1,2,\dots. $$ I want to study some asymptotic results of $f_n$. For ...
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49 views

$X_n$ doesn't converge to a limit in $[-\infty, \infty] \to$ Is this supposed to be a stronger version of $\lim X_n$ doesn't exist?

From Williams' Probability with Martingales: What's the difference between saying that '$X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$' and '$\lim X_n$ does not exist' ? ...
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2answers
18 views

Using the definition of the convergence of the sequence, how can I prove that this sequence converges to the limit?

I suppose what I am confused most on here is the algebraic method. $\lim_{x\to\infty} $ $ 2n^2/(n^3 + 3) $ = 0 I have set it up so far: Let $ \epsilon > 0 $ be given. I set up the equation ...
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2answers
57 views

How to tell if the series $\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4}$ converges?

$$\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4} $$ Trying to figure out if this converges, trying to use the divergence test but I can't figure out how to simplify the problem.
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1answer
22 views

Sum of Specific Convergent Series

Let L be the sum of the following alternating convergent series $$L = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}$$ Now consider the rearrangement ...
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0answers
27 views

Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
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0answers
36 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
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1answer
40 views

Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly

$$X=\{ \alpha:[a,b] \rightarrow \mathbb{R} \}$$ $\alpha''$ exists and it is continuous $$\exists \ K>0 \ : \forall \ x \in [a,b], \forall \alpha \in X: \\ \ \\ \rvert \alpha(x) \rvert, \rvert ...
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22 views

Show that $Z_n$ converges to $0$ in mean square

Let $Z_n$ be a discrete random variable with mass function $\mathbb{P}(Z_n = n^\alpha) = \frac{1}{n}$ and $\mathbb{P}(Z_n = 0 ) = 1 - \frac{1}{n}. $ I want to show that $Z_n$ converges to $0$ in ...
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1answer
20 views

Why is $\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolut convergent?

Why is $\sin(tA)=\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolute convergent ? for a $n\times n$ real matrix $A$ and $t\in \mathbf R$ Which crieterion is to use ?
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1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
3
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1answer
34 views

Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$

I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$. The ...
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4answers
95 views

How to prove that $\sum_{n=1}^{\infty} \frac{(log (n))^2}{n^2}$ converges?

$$\sum_{n=1}^{\infty} \frac{(log (n))^2}{n^2}$$ I know that this series converges (proof by Answer Sheet). However I need to prove it using comparison, integration, ratio or other tests. The ...
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1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
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24 views

Compute the order, type and genus of the entire function $\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}z \right) $

Since $$\sum_{n=1}^\infty\frac{1}{(n^3/\sigma(n))}=\frac{\pi^2}{6}\zeta(3)$$ converges, where $\sigma(n)$ is the sum of divisor function (with maximal size a constant times $n\log\log n$), and ...
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2answers
86 views

Decide whether the series ${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$ converges or diverges

Determine whether the series converges or diverges $${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$$ I was thinking I should use ratio test but I get an ugly sequence that I don't know how to ...
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2answers
22 views

How do I find explicit formula for $(x_n,y_n)$ and show that the sequence tends to converge to $(0,0)$?

How do I find explicit formula for $(x_n,y_n)$ and show that the sequence tends to converge to $(0,0)$? In $\mathbb{R}^2$, the sequence $(x_n,y_n)$, $n\in \mathbb{N}_0$ is recursively defined: ...
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1answer
31 views

Convergence of a sequence of roots of continous functions

Let $(f^n,n\in\mathbb{N})$ be a sequence of complex continous functions so that $f^n(u)\longrightarrow f(u)$ uniformly to a complex continous function f if $n \longrightarrow \infty$. I addition I ...
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1answer
37 views

Counterexample that a measurable function does not guarantee almost sure convergence

I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$ ...
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1answer
32 views

Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent

I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that ...
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1answer
105 views

Convergent & Cauchy Sequence related prove

(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $$\{a_n\}\to a$$ and $$\{b_n\}\to b$$ for $n\to\infty$. Prove that $$\{a_n+b_n\}\to a + b$$ for $n\to\infty$. (2) Prove ...
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1answer
33 views

Bound for series

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
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2answers
50 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
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0answers
45 views

Summation Involving Hermite Polynomials

From the generating formula for Hermite polynomials we know that $$ e^{2xz - z^2} = \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n!} \, . $$ The sum $$ \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n! ...
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25 views

Bounding of series

Please help - I tried writing the LHS term as a sum and was hoping to find a 'known' convergent sequence that would bound it, but haven't had any luck. Let R > 1. Show that there is some M > 0 such ...
2
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18 views

Bound for series in Gamma proof

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
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2answers
40 views

Convergence of $\sum \sin\frac{(-1)^n}{n^p}$

$$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p}\quad p>1$$ My attempt: $$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p} = \sum_{n=1}^{\infty} (-1)^n\sin\frac{1}{n^p} $$ And $\sum_{n=1}^{\infty} ...
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1answer
36 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
2
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2answers
74 views

$L_p$ space and convergence

Let $f_i\rightarrow f$ $m$-a.e on $[0,1]$, $m$ is a measure and $\int|f_i(x)|^4dm$$\le1$ for all $i$.Then $\int|f_i(x)|^2dm\rightarrow \int|f(x)|^2dm$. how to prove it? in my solution i prove that ...
3
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1answer
24 views

Lebesgue integration and uniform convergence

Let $\Omega$ be a bounded and measurable set in $\mathbb{R}$. If $\{f_n\}$ is a sequence of bounded and Lebesgue integrable functions on $\Omega$. If $f_n$ uniformly converges to $f$, then how to ...
2
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1answer
151 views

Convergence in distribution then in probability

If $X_n$ $\rightarrow$ $c$ in distribution, then $X_n$ $\rightarrow$ $c$ in probability. How do I show this? I thought convergence in probability implied convergence in distribution?
2
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2answers
19 views

Check convergence and sum of a sum of finite sum.

$$\sum_{n=1}^\infty \sum_{k=1}^m \left(\frac{x_k}{y}\right)^n\quad 0<x_k<y$$ My attempt: Convergence: Since $\frac{x_k}{y} <1$ we can conclude that: $$\sum_{k=1}^m ...
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2answers
22 views

Designing a Power Series with certain $R$

Out of interest, is there a way to design a series with a certain radius of convergence? For example, $R=8$, or is there a way to turn a series for which the Radius of Convergence is known, then ...