Convergence of sequences and different modes of convergence.

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1answer
66 views

how can I find the convergence of the integral $\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx$ , for $ x \in [0,1]$ [closed]

I want to check the convergence of the integral $$\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2} dx $$ for $ x \in [0,1]$ and n->∞ is a constant so can basically pulled out of the integral but I don't know ...
1
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3answers
67 views

how can I find the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , $ x \in (-1,1)$ [closed]

I want to check the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , for $ x \in (-1,1)$ but i don't know what to do. Every theory I know it is not working. Can someone ...
0
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1answer
27 views

Radius of Convergence on Power Series Help

I am struggling to find the radii of convergence of the following two series: $$\sum_{n}n^{\cos(n)}z^n$$ $$\sum_{n}(2^{-n} + 3^{-n})z^n$$ Here I tried using ratio test and lim sup, but didn't ...
1
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1answer
38 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in L^...
0
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0answers
30 views

Is this a rigorous enough argument against uniform convergence?

I have $\sum_{n=1}^{\infty} \frac{ln(1+x/n)}{n}$ and I need to prove that it does not converge uniformly on $[0,\infty)$. So I use the delta-epsilon definition of convergence and then I expand this ...
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1answer
31 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
2
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1answer
83 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

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 ...
0
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0answers
41 views

Convergence of sequence of vectors in $C^*$-algebra

Let $B$ is $C^*$-algebra and $x_i \in B$ - linear independent vector system, $\alpha_i \in \mathbb{C}$ such that: $$\|x_i\| = 1$$ $$\lim_{N \to \infty} \|\sum_{k=1}^N \alpha_k x_k\| = \lim_{N \to \...
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0answers
46 views

Evaluating convergence of a non-trivial series

I am trying to evaluate whether the following limit is finite (as opposed to being $\infty$): $$\lim_{n \to \infty} \sum_{k=2}^n \frac{1}{n-1} \left \{\sum_{i=2}^l (i-1)\frac{(n-i)!}{(n-i-k+2)!} \...
3
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2answers
68 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
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2answers
107 views

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 ...
0
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3answers
76 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 $$\frac{1}{2}\int_0^...
0
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1answer
21 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 ...
0
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1answer
15 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 ...
2
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1answer
29 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
34 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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0answers
18 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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0answers
16 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 $\Rightarrow$ $\int_{0}^{\...
3
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1answer
67 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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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 ...
0
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2answers
69 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)] = \frac{1}{2}\log\left[\left(1+\frac{1}{2^{(n+1)}}\right)^{2^{n+1}}\...
4
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1answer
57 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.
3
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2answers
51 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.
2
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1answer
36 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 ...
0
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2answers
32 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: $f(n)=1/(n-1)$ ...
7
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1answer
147 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 \sin{...
3
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0answers
24 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 $P^{...
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45 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 $p_n = \sum_{i=1}^n \alpha_i e_i$...
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1answer
71 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 ...
0
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1answer
17 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 $\{...
0
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1answer
38 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 $p_n = \sum_{i=1}^n \alpha_i e_i$...
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1answer
35 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
51 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 $$x_k = 1 - 2^{-k}$$...
3
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3answers
146 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
78 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 ...
0
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1answer
50 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 ...
2
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1answer
23 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 $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin nx.$$...
2
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2answers
39 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
30 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 =...
1
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1answer
68 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 ...
3
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1answer
26 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 ...
2
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3answers
75 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 $x \...
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0answers
22 views

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 + \...
-1
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1answer
17 views

Bounded convergence theorem - 2M

Can someone please help me with where the 2M is coming from?
4
votes
3answers
83 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 $H_x$ ...
0
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0answers
31 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 ...
0
votes
1answer
38 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 ...
1
vote
2answers
34 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 \...
0
votes
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 $$g\left(x\right)=2\sum^{\infty}_{n=1}\dfrac{\left(-1\right)^{n+1}}{n}...
0
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2answers
33 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?