Convergence of sequences and different modes of convergence.

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Prove convergence of sequence

A sequence $\{x_n\}$ satisfies $$|x_{n+1}-x_n| \leq \alpha|x_n-x_{n-1} | $$ for $n=2,3,4,\ldots$ and the number $ 0<\alpha<1$ fixed. Prove $\{x_n\}$ is a convergent sequence.
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74 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
34 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
44 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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66 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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1answer
37 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
66 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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10 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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15 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
58 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
53 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
170 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
30 views

Discuss the convergence of $\sum a^n/(x^n+a^n)$?

I fail to understand how any of the tests I know will work on this. I tried the D'Alembert's Ratio test but I don't understand how it will limit to something.
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1answer
51 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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43 views

Inner Product Properties And Applications

In every calculus or analysis class we are told that the concept of inner product is very important, and that its applications are vast, diverse, and extremely useful. I don't think there is a single ...
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1answer
32 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
19 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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26 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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2answers
45 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;when\;|\omega|\to\infty$$ I get all the steps of the proof, except the one ...
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1answer
69 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?
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1answer
57 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
36 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
85 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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2answers
41 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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4answers
393 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
46 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis, 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, for ...
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4answers
88 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
35 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
41 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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1answer
91 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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2answers
36 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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51 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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28 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
52 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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20 views

Almost sure uniform convergence [on hold]

If a sequence of continuous random variables converges almost surely uniform to a random variable, is it true that the limit is $\mathbb{P}$-almos surely continuous?
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1answer
81 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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31 views

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

$\lim_{n \rightarrow \infty} sup_{\delta \rightarrow 0} (n \delta)^{-1} |T_n(\theta)_{ij}| < \infty$ a.s, $1 \leq i \leq p$, $1\leq j \leq p$ what does the multiplication of $n$ and $delta$ mean, ...
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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
46 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 ...
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1answer
25 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 ...
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24 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 ...
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1answer
70 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 ...
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1answer
45 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
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1answer
29 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
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4answers
54 views

How do I determine $r$ in this geometric series $a+ar+ar^2+\cdots$? [closed]

The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of 5. What is $r$? Thank you for any help .
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3answers
34 views

Convergence of Sequence Proof Question

I have just learned about the convergence of a sequence with the epsilon definition. So when we try to prove a limit of the sequence, what are we doing essentially (with respect to the definition)? ...
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3answers
83 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
0
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0answers
42 views

Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...