Convergence of sequences and different modes of convergence.

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If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
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39 views

Evaluate $\int_0^1y ( ( 1+\frac{1}{y^2} )\log (1+y^2) -1 )dy=-1+\frac{\pi^2}{24}+\log 2$ and a related generalization

Let $0<x<1$ and $0<y<1$ thus $\xi=xy^2<1$ and we can use the series expansion $$\frac{1}{2}\log\frac{1+\xi}{1-\xi}=\sum_{n=0}^\infty\frac{\xi^{2n+1}}{2n+1}$$ to get ...
3
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0answers
20 views

$\langle w,\varphi\rangle =\int_{\mathbb{S}^1} \left(\sum_{k \geq 1} e^{itk}\right) \varphi(t) \, dt$ - Generalized function

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
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1answer
13 views

Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is ...
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1answer
12 views

$<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ know that $\varphi(0)=0$ - Generalized function

Question : Show that $<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ for any $\varphi \in D(\mathbb{R})$ for which $\varphi(0)=0$. I am a little bit confused how to solve ...
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1answer
23 views

$\lim_{n \to \infty} \langle f_n, \varphi \rangle$ - Generalized function

Question : Let $f_n$ be the distribution $<f_n,\varphi>=n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n}))$. What distribution is $\lim_{n \to \infty} <f_n, \varphi>$ ? First try : $\lim_{n ...
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1answer
28 views

The sum of series in an interval

I have the following series - $$ \sum_{n=1}^\infty nx^{2n-1} $$ I found that its convergence interval is $[-1,1]$ but how can i calculate the sum in this interval ? i would like to get some hint ...
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14 views

convergence of $\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)$

The identity $$\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)\qquad (1)$$ is well-known and valid for $s\in\mathbb{R}$ with $s>\max\{1,1+k\}$. $\sigma_k$ is the divisor function. ...
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8 views

Proof: Majorant- and minorantcriterion for convergence of improper integrals

Let $I$ a interval and let $f,g:I\to \mathbb{R}$ continuous with $0\le g(x) \le f(x) \forall x \in I$. Prove this propostitions: (a) If $\int_{0}^{\infty} f(x)dx$ convergent $\Rightarrow$ ...
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1answer
53 views

Determining convergence of a series $\sum_n (-1)^n \sin a_n $

I need to determine if the following series is convergent: $$\sum_{n=2}^\infty (-1)^n\sin\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n\ln^2(n)}\right).$$ I've tried to use alternating series test but ...
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1answer
20 views

Let {xn} be a sequence that has two subsequences converging to different limits. Prove {xn} is not convergent. [on hold]

I can't use that if {xn} is a convergent sequence, then any subsequence {xni} is also convergent and their limits are equal.
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0answers
16 views

Finding where a complex series converges absolutely, uniformly.

I need to figure out where the series converges absolutely and uniformly. I know that once I have absolute convergence on a region, then I know I also have uniform convergence on that region, so I ...
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2answers
64 views

Proving that the series $\sum [\log(2n+1)-\log(2n)]$ diverges.

Let $f(n)=\log(2n+1)-\log(2n)$. Using the Cauchy's condensation test we have: $$2^nf(2^n)=2^n[\log(2\cdot2^n+1)-\log(2\cdot2^n)] = ...
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1answer
55 views

Check series for convergence

$$ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)^n}{n^{1/4}}\right) $$ I have no idea how to deal with it.
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2answers
47 views

Check for convergence

$$\sum_{n = 2}^\infty (-1)^n \sin\left( \frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n (\ln n)^2}\right)$$ I tried to use Maclaurin series, but failed to evaluate little-o.
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1answer
31 views

Using MCT twice to show the limit of an integral depending on $x$ and $n$

So I have $\displaystyle\lim_{n \to \infty} \int^{n^2}_0 e^{-x^2} n \sin\left(\frac{x}{n}\right) dx$. I'd like to apply the MCT but the trouble is there is a limit which also depends on $n$ So I ...
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2answers
29 views

Convergent sequences must have bounded range?

I am currently reading Baby Rudin and I am having trouble understanding why convergent sequences must have a bounded range. Specifically, I am thinking of the following counterexample: $f(n)=1/(n-1)$ ...
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39 views

Series Convergence by Comparison [on hold]

I've been working on some calculus problems and am struggling to understand what I can compare this series to in order to prove convergence: $$ \sum_{n=0}^{\infty} \sin^2(\pi n) $$ I know that all ...
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0answers
16 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
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1answer
137 views

Prove that sum is convergent

How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$ I tried to use formula $$\sin(n+ \ln{n}) = \sin{n}\cos \ln{n} + \sin \ln{n}\cos{n}$$ and $$\sum_1^N ...
3
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0answers
15 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures ...
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0answers
40 views

About сonvergence of partial sums of basis of Banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i ...
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1answer
66 views

Proving that $\lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$

Prove that $\displaystyle \lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$ I was trying to proof in the same way of $\lim (1 + \frac{1}{n})^n = e$, but I couldn't proceed this way. Can someone give me a ...
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1answer
15 views

Uniform convergence of supremum

If a sequence $\{f_n\}$ converges uniformly to a limit $f$ on the domain $D$, then the sequence $\{M_n\}$, with $M_n = \sup_{x} |f_n(x)-f(x)| $, converges to zero. So what I thought was since ...
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1answer
37 views

Convergence of partial sums of basis vectors in banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i ...
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1answer
30 views

Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

I've tried to partial this integral from 0 to 1, 1 to e, and e to infinity. $$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx$$
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3answers
46 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...
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3answers
89 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
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3answers
73 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
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1answer
48 views

Are these integrals convergent?

Recently I've come across two integrals that seemed hard to check for me. Here they are: $$\int_0^\infty \frac{x \sin \ln x}{x^2 + \cos x} \, \mathrm{d}x$$ And another: $$\int_1^\infty \frac{\sin \ln ...
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1answer
22 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
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2answers
36 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
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3answers
29 views

Proving a subsequence doesn't converge

When I want to prove that a sequence doesn't converge by showing that it's subsequence doesn't converge , can i use the limit comparison test? (Usually used for series) . for example - $$ \sum_{n ...
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1answer
65 views

Convergence of $g(x)\cdot f(x)$

Let $g(x)=\frac{1-e^{-x^2}}{x^2}$ for $x \neq 0$,$g(0)=1$ and $f(x)=e^{-(x-n)^2}$. You can assume that g(x) is continuous and bounded with maximum 1 in x=$0$. Show that $\sum_{n=1}^{\infty}g(x)\cdot ...
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1answer
21 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
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70 views

How to show $\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$ does not exist?

$$\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$$ I my intuition is telling me this limit does not exist as $\sin$ will be oscillating but will stay bounded and then will blow up as ...
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Examples 3.35 (a) and (b) in Baby Rudin: Limit Superior and limit inferior of a couple of sequences

This question is related to Examples 3.35 (a) and (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, p. 67. Let us consider the series $$ \frac 1 2 + \frac 1 3 + ...
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1answer
17 views

Bounded convergence theorem - 2M

Can someone please help me with where the 2M is coming from?
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3answers
78 views

What is $\lim_{x\to \infty} 2\sqrt{x}- \sum_{n=1}^x {1\over \sqrt{n}}$? [duplicate]

I ask this because I noticed the partial sum $\sum_{n=1}^x {1\over \sqrt{n}}$ is very close to $2\sqrt{x}$, so close in fact that it appears their difference approaches a constant value, like $H_x$ ...
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0answers
29 views

Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
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1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
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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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0answers
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} ...
0
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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, ...
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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2answers
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.
0
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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 ...
0
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1answer
42 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 ...