Convergence of sequences and different modes of convergence.

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20 views

Are these partial sums and partial products absolutely convergent?

For arbitrary $m \in \mathbb{N},$ $$\sum_{n=1}^{m}\ \sum_{d | \#_n}\mu(d)=\sum_{n=1}^{m}\big | \sum_{d | \#_n}\mu(d)\ \big |\ = \ 0,$$ $$\prod_{n=1}^{m}\ \prod_{d | \#_n}d^{\mu(d)}=\prod_{n=1}^{m}\big ...
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1answer
45 views

Why are higher order derivatives linked to the higher orders of convergence?

My textbooks defined the rider of convergence as follows (original image link) For an iterative process of the form $x_{n+1} = g(x_n)$, the order of convergence is first order when $|g'(x)| ...
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4answers
61 views

What is the correct radius of convergence for $\ln(1+x)$?

My text tells me this: And, Wolfram tells me this: Now, I'm not certain what to believe, but I believe I'm not certain because I'm not certain if Wolfram is using the logarithm with base $10$. ...
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2answers
56 views

The limit of a composed function

Let $g_n: \mathbb{N} \rightarrow \mathbb{R}$ and $f_n(x): \mathbb{N\times R} \rightarrow \mathbb{R}$. If $g_n \rightarrow g$ and $f_n(g) \rightarrow f(g)$, can we deduce $f_n(g_n) \rightarrow f(g)$? ...
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1answer
125 views

For which values does this series converge?

p and k are real numbers. For which values of p and k does the following double series converge $$\sum_{n,m=1}^\infty \frac{1}{n^p + m^k}$$ I am trying to find a better (and quicker) way to solve ...
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2answers
25 views

Looking for example of point wise convergent continuous bounded sequence of functions whose limit is neither continuous nor bounded

I am looking for a sequence of real valued functions $\{f_n(x)\}$ with domain some subset of $\mathbb R$ such that each $f_n$ is bounded , continuous and $f_n$ converges point-wise to some function ...
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28 views

How do I prove that $f(s)=\sum a_{n}{n}^{− s}$ converge for $Re(s)>0$if the partial sum of $a_{n}$ are bounded?? [duplicate]

let $f(s)$ be a power series defined as follow :$$f(s)=\sum a_{n}{n}^{− s}$$ Assume the partial sum of $a_{n}$ are bounded .My question here is : How do I prove that $f(s)$ converge for ...
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1answer
20 views

Radius of convergence of a power series (little question about power + constant)

My power series are: $$\sum _{n=1}^{\infty }\:\frac{x^{3n+1}}{\left(1+\frac{1}{n}\right)^{n^2}}$$ So its isnt difficult if it was written without the $+1$ in the power: $$\sum _{n=1}^{\infty ...
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37 views

help with investigating the uniformly convergence of a function sequence.

I have to check the uniform convergence of the below mentioned function sequence: $f_n(x) = \frac{1-\ln x}{nx}$ while $0<x<1$ at the answers, it's told that the sequence doesn't converge ...
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72 views

Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
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3answers
87 views

Determining the value to which the sequence $a_n=\frac{n!}{n^n}$ converges.

How can it be deduced that the sequence $a_n=\dfrac{n!}{n^n}$ converges to $0$? I can reasonably infer this to be true, because I see the pattern as $n$ approaches larger values, but I am unsure of ...
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1answer
40 views

Evaluating the convergence of the sequence $\{a_n\}=\frac{(-1)^{n-1}n}{n^2+2}$.

Set the sequence $a_n$ such that $\{a_n\}=\dfrac{(-1)^{n-1}n}{n^2+2}$. If $|a_n|$ converges (only to $0$, it would seem; correct me if I'm wrong), then $a_n$ must too converge, both to some value $L = ...
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0answers
48 views

Is this $\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $ alternating series for all values of $\theta$?

I have tried to do other form of alternating series I got this: $$\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $$ Can I say that the above series is alternating series for all ...
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2answers
80 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
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49 views

Determine whether $\sum \frac{1}{n^3 \ln(n^4+9)}$ converges

For the series $$\sum_{n=2}^{\infty}\dfrac{1}{n^3 \ln(n^4+9)},$$ I was thinking of using the limit comparison test with $1/n^3$?
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4answers
74 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{(3n+8)!}$

Determine whether or not this series converges or diverges $$\sum_{n=1}^{\infty}\dfrac{1}{(3n+8)!}$$ My attempt: I used the ratio test and ended up having ...
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1answer
50 views

Determine whether the series converges or diverges?

For the series $$\sum_{n=3}^{\infty}\dfrac{3n^2+8n}{7n^3-4n^2+11},$$ I was thinking of using the limit comparison test to $\dfrac{1}{n}$ but is there a better way?
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1answer
17 views

Can I show that a process which a supermartingale above a certain value and a submartingale below it converges?

In my work, I have many times encountered dynamic stochastic systems which are a submartingale (increasing in expectation) below a certain value of the variable, $X^*$ and a supermartingale ...
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0answers
18 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
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1answer
63 views

what is the radius of convergence of power series $\frac{z^2}{z}$?

I have a power series and am being asked to find its radius of convergence, but its structure of type $$\sum\frac{z^{2n}}{z^n}$$ is confusing me. How do I calculate radius of convergence of this power ...
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2answers
59 views

Prove convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$

Prove the convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$ This was a question on an exam. I needed to prove that the above integral converges using the comparison test. I thought about ...
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2answers
172 views

Help with Convergence of a series with sin and log

I tried to figured it out if the follwing series converges or not $$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\ln^2n}\ (-1)^{n}$$ I tried to show that $\sin(\frac{1}{n})$ is a monotonic but I'm ...
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1answer
52 views

Converging integral $\int_1^\infty {\frac{\sqrt{x}\cos{x}}{x+2013}}dx$

I want to show that $$\int_1^\infty {\frac{\sqrt{x}\cos{x}}{x+2013}}dx$$ is converging. I tried $${\frac{\sqrt{x}\cos{x}}{x+2013}}\leq {\frac{\sqrt{x}\cos{x}}{x}}\leq \frac{1}{\sqrt{x}}$$ but it ...
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1answer
48 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow ...
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3answers
22 views

Radius of Convergence

Is the radius of convergence of $$\frac{n(x+3)^n}{4^n}$$ equals 4? I got $|x+3|\lt 4$ as the final result. How do you know, what is the radius from here?
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41 views

Should I use the comparison test for the following series?

Given the following series $$\sum_{k=0}^\infty \frac{\sin 2k}{1+2^k}$$ I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would ...
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58 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
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1answer
70 views

Using the ratio test when the limit of ratio is infinity

If the limit in ratio test is infinity. Does the sequence converge? I suspect not as it is infinity and not some finite value but I'm not sure. Any help?
3
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1answer
59 views

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, ...
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1answer
39 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
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2answers
94 views

Two divergent series such that their product is convergent

I faced a series question it goes something like give an example of 2 divergent series such that when the 2 series are multiplied to each other, the new series becomes convergent, although it looks ...
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51 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
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405 views

Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
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1answer
73 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis I, specifically the sequence and series part. I've just come across the definition of convergence: A sequence $(a_n)$ converges to a real number $a$ if, ...
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89 views

If $x_{n+1}\leq x_n + 1/n^2$ then $x_n$ converges [duplicate]

Let $x_n$ be a sequence of non-negative real numbers such that $\forall n, x_{n+1}\leq x_n+ \frac{1}{n^2}$ Prove that $x_n$ converges. The problem is trivial whenever $x_{n}$ is an ...
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1answer
37 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
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2answers
42 views

Struggling with the integrability of $\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$

I know quite a lot tools to determine the integrability of functions, but in this case I really don't know where to start: $$\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$$
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96 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
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39 views

How can I solve this integral with the comparison theorem?

I have an integral that I am not sure how to solve with the comparison theorem to see if it is divergent or convergent. $$\int_1^\infty\frac{e^{-2x}}{\sqrt{x+16}}\;dx$$ How can I solve this with ...
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2answers
63 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
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34 views

suppose $a_n>1$ $a_n$is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}(1-\frac{a_n}{a_{n+1}})\frac{1}{\sqrt{a_{n+1}}}$ converges

suppose $a_n>1$, $\{a_n\}$ is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}\left(1-\frac{a_n}{a_{n+1}}\right)\frac{1}{\sqrt{a_{n+1}}}$ converges I don't have any idea about how to ...
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3answers
54 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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1answer
97 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
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31 views

Limit of Convergent Sequence Property Proof Help

I have a question about this property: Let $\lim\limits_{n\to\infty} a_n = a$, then $\lim\limits_{n\to\infty}(ca_n) = ca$ for all $c \in \mathbb R$ If we consider when $c$ doesnt equal $0$, my ...
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0answers
38 views

what does this mean $n\delta$ where $n\rightarrow \infty$ and $\delta \rightarrow 0$

$\lim_\limits{n \rightarrow \infty} \sup\limits_{\delta \rightarrow 0} \left(n \delta\right)^{-1} \left|T_n(\theta)_{ij}\right| < \infty$ a.s, $1 \leq i \leq p$, $1\leq j \leq p$ what does the ...
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3answers
56 views

How can i prove that this integral is convergent/divergent

This is my equation: $$\int_0^{\pi/4} \frac{dx}{x\sin2x}$$ I wish to prove that it's convergent or divergent, by $P$ test and/or comparison test, but it does not seem to be applicable... Is it ...
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3answers
48 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
1
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1answer
28 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
2
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0answers
26 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
3
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1answer
73 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that ...