Convergence of sequences and different modes of convergence.

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2
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2answers
52 views

Prove sequence $S_n$ converges

If $S_1 = \sqrt{2}$, and $S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....), prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$ This is one the questions ...
1
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5answers
76 views

Does the series $\sum_{n=1}^\infty \frac{n+1}{n^3+10n}$ converge?

Using the ratio test, we evaluate: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty}\left| \frac{(n+1) + 1}{(n+1)^3 + 10(n+1)} \cdot \frac{n^3+10n}{n+1} \right| = \lim_{n ...
0
votes
3answers
38 views

Prove sequence $\frac{5n+1}{n^{5}-2}$ is convergent

Please can someone help prove that the sequence $\frac{5n+1}{n^{5}-2}$ is convergent from first principles? Thanks
1
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2answers
32 views

sequence ${1/n}$ converges in R (to 0) but fails to converge in the set of all positive real number. Why?

Sequence $\{\frac{1}n\}$ converges in $\mathbb{R}$ (to $0$) but fails to converge in the set of all positive real numbers. I don't fully understand why it fails to converge in the set of all positive ...
0
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2answers
22 views

Convergence of $\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n} $

I need to prove that the following series converges $$\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n}$$ for $|q| < 1$, $|z| < 1$. I thought maybe to use some comparison tests, but I'm not sure if ...
0
votes
2answers
30 views

Real Analysis: Limit Proof with help of ratio convergence property

The thing is to, using the property above, prove the expression below. First tried to obtain the difference between n and n+1 terms, then factored out n+1 ^ p and rearanged the sequence. I've ...
2
votes
3answers
50 views

Absolute convergence of $\sum_{n=1}^{\infty} (-1)^n (1-\cos1/n)$

I would like to see if $$\sum_{n=1}^{\infty} (-1)^n \left(1-\cos \frac{1}{n}\right)$$ converges absolutely.
3
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4answers
49 views

$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$convergent/divergent?

Please could someone help prove $$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$$ converges/diverges? Thank you.
0
votes
2answers
17 views

Convergence of this alternating series?

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
-1
votes
1answer
38 views

If limn→∞(an − bn) = 0 and sum to infinity (bn) converges, then sum to infinity (an) converges. True or false? [closed]

If $\lim_{n\to\infty}(a_n − b_n) = 0$ and $\sum b_n$ converges, then $\sum a_n$ converges. I need to prove whether this is true or false, and if it is false can anyone please give a counter example? ...
0
votes
0answers
27 views

Calculation of value a series converges to [closed]

$$\sum_{k=0}^n\dfrac1{{}^{n+k}C_n}.$$ How can we calculate the value the series converges to for different values of $n$?
5
votes
3answers
136 views

How to calculate convergence of this function?

Given $n$, is there any easy way to calculate convergence of this summation. $$\sum_{k=0}^\infty\dfrac{1}{^{n+k}C_n}$$ EDIT: Also I need to find at which value this series converges.
3
votes
1answer
57 views

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

I'm interested in finding a closed form for $$\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$$ Wolfram Alpha confirms that it converges, but I can't find any plausible closed forms. I've made ...
2
votes
3answers
69 views

Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
1
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0answers
21 views

Practical Intuition behind converge of random sequence $X_n \to X$

Can someone come up with a practical example as to what it means for a set of random variables to converge i.e. $X_n(\zeta) \to X(\zeta)$? Specifically, what is the meaning of $n$ and $\zeta$?
4
votes
1answer
92 views

Why does my money converge to zero?

I invest $\$1000$ in a stock. Every year, the stock price will either increase by $90\%$ or decrease by $50\%$. The expected value of the change in the price is: $0.5\cdot90\% + 0.5\cdot(-50\%)= 20\% ...
0
votes
0answers
24 views

Convergence of random variables depending on the measure

Suppose the probability spaces $\left([0,1], \mathcal{B}([0,1],\mu_i \right)$ for $i=1,2,3$ , where $$ \mu_1 = \lambda , \ \ \ \ \ \ \ \ \ \ \mu_2 = \delta_1,\ \ \ \ \ \ \ \ \ \ \mu_3 = ...
1
vote
1answer
36 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} ln(2+x)dx$$ Don't know where to start..
3
votes
2answers
97 views

How to show that $\sin(n)$ does not converge?

I am supposed to show that $\sin(n)$ does not converge by constructing two subsequences: one subsequence contains terms of $\sin(n)$ that are between $1/2$ and $1$, and the other subsequence contains ...
1
vote
1answer
31 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...
1
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0answers
43 views

Clarification: Rudin Theorem 3.7: Subsequential limits are closed

My question is this: Why does Rudin use $\delta$ in this proof? Would it not work just as well if $\forall i \ge1,$ $$x_{i}\in N_{2^{-(i+1)}}(q) \cap E^* $$ $$p_{n_i}\in N_{2^{-(i+1)}}(x_i) ...
3
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2answers
25 views

Show convergence in distribution by the continuity theorem

So the problem I'm about to solve is to show that: $X \in \Gamma(a,b)$. Show that \begin{equation} \frac{X-E[X]}{\sqrt{Var(X)}} \xrightarrow{d} N(0,1) \end{equation} as $a \rightarrow \infty$, by ...
0
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0answers
25 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
0
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1answer
26 views

dense subspace of $L^2(\Omega\times(0,T))$

I am trying to prove that the functions $f(\omega,t)=g(\omega)h(t)$ where $g\in G,\: h\in H,$ are dense in $L^2(\Omega\times(0,T))$ if $G$ is dense in $L^2(\Omega)$ and $H$ is dense in $L^2((0,T))$. ...
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votes
2answers
40 views

Is there an easy way to check convergence of real sequence? [closed]

How to check convergence of a sequence (for example, of $x_n =[nx]/n$ for a fixed $x \in \mathbb R$.) For series we can use various tests to check its convergence which are not available for ...
0
votes
0answers
13 views

Understanding $O_p$ [migrated]

One thing I feel like I have never mastered is the concept of $O_p$ convergence and how to use it. I understand the basic idea and what bounded in probability means, but I always have a hard time ...
1
vote
2answers
29 views

Property of Sequence Dense in $[0,1]$

Suppose $\{x_i\}_{i = 1}^{\infty}$ is a dense sequence in the $[0,1]$ interval. For every $\epsilon > 0$, does there always exist an $n$ such that $|x_n-x_{n+1}| < \epsilon$?
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2answers
52 views

Sum of the distance between consecutive terms implies the sequence is Cauchy.

If a sequence $(x_n)_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum_{n\geq 1} ||x_n-x_{n+1}||<\infty$, show that it is Cauchy. This isn't a complete answer, but here's my train of thought. ...
0
votes
1answer
36 views

Cesaro summability and $\sum n \lvert a_n\rvert ^2 < \infty$ implies convergence

How can I prove that if $\sum_{n=1}^\infty a_n$ is Cesaro summable and if $\sum_{n=1}^\infty n |a_n|^2 < \infty$, then $\sum_{n=1}^\infty a_n$ converges?
0
votes
1answer
45 views

Seeking a possible counterexample in probability.

I am trying to find a counterexample or prove the following: $\dfrac{Var\left(X_{n}\right)}{\left[EX_{n}\right]^{2}}\rightarrow0 , then \dfrac{X_{n}}{EX_{n}}\rightarrow1$ in probability. Assuming ...
0
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1answer
14 views

convergence of this series with n^a*(\log(n))^b in denominator

How do I proceed with finding a and b such that $\sum_{n=2}^{\infty}\frac{1}{n^a {(\log(n)})^b}$ converges ? Which test is the most appropriate to use and find the values ?
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0answers
8 views

Z-transform and región of convergence (ROC) [duplicate]

I need complete this problem. Any help me?
0
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2answers
21 views

convergence and sum of this series

$\sum_{n=1}^{\infty}(\frac{1}{n}-ln(1+\frac{1}{n}))$ is supposed to be convergent. If I use the integral test, I can prove the second term to be a finite integral while the first term is still ...
10
votes
1answer
74 views

Does the series $\sum_{n=1}^{\infty}\sin\left(2\pi\sqrt{n^2+\alpha^2\sin n+(-1)^n}\right)$ converge?

Let $\alpha$ be such that $0\leq \alpha \leq 1$. Since $\sin n$ has no limit as $n$ tends to $\infty$, I'm having trouble with finding if the series $$\sum_{n=1}^{\infty}\sin ...
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2answers
146 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
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1answer
29 views
0
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1answer
17 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
3
votes
0answers
33 views

Does $L^{1}$ convergence implies almost everywhere cesaro convergence?

Let $X$ be a compact metric space, $\mu$ a Borel measure and $f_{n} \in L^{1}(X,\mu)$ continuous functions such that $f_{n}(x) \overset {L^{1}} \rightarrow 0 $. Now can we deduce that ...
0
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1answer
41 views

$P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$ for all $a < b$ implies that $lim_{n \rightarrow \infty} X_n$ exists a.e.

Suppose for $a<b$ we have $P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$. Then $lim_{n \rightarrow \infty} X_n$ exists a.e. but may be infinite. Here "i.o." means "infinitely often"; for any ...
0
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2answers
26 views

Monotone and bounded or just bounded

I have to know if is it necessary that a sequence to be bounded and monotone to have a limit or it can be just bounded? For example, $A_{n+1}=1/(1+A_n)$, it is bounded but not monotone. However it has ...
0
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1answer
19 views

Essential Uniform Convergence Implication

Greetings Mathematics Community. I believe that I am thinking too hard about the following problem and would like some guidance in solving it. Let $X$ have finite measure and let $f_n:X \to ...
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2answers
24 views

study of two sequences

I need to study wether thos two sequences converge or not. 1) $u_n=n\sum_{k=1}^{2n+1} \frac{1}{n^2+k}$ 2) $v_n=\frac{1}{n}\sum_{k=0}^{n-1} \cos(\frac{1}{\sqrt{n+k}})$ For the first i get it ...
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1answer
26 views

Real number approximation by a product of two integer powers

Inspired by the question on which points of the $\mathbb{C}$ unit circle can be reached by arbitrary products of two example points, I came up wtih the following: For given $a, b \in \mathbb{R}^+ ...
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0answers
28 views

Trying to find sum of a complex infinite series with Gamma function and factorials

I am trying to find the sum $S$ of the following series. $$S = \sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}t^{2n}\Gamma\left(\frac{1 + 2nH - ...
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0answers
23 views

Is my convergence proof correct?

It's "obvious" that the following sequence converges. I was asked to prove it on a homework assignment and was given no credit for my proof. I wanted to ask the community here if 1) they think my ...
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2answers
26 views

Show $\sum_n \left(1-\frac{K}{n^{1-\epsilon}\sqrt{\log n}} \right)^n$ converges for $\epsilon>0$.

This is not a homework problem. It has come up in my research. I am trying to show that $$\sum_n \left(1-\frac{K}{n^{1-\epsilon}\sqrt{\log n}} \right)^n$$ converges for $\epsilon>0$. I have no ...
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2answers
44 views

The convergence of an infinite seqeunce

Suppose that $$ a_n = \prod_{k=n}^{\infty}\left(1 - \frac{1}{k^2}\right), $$ for $n \geq 2$. How can we show that $$ \lim_{n \to \infty} a_n = \lim_{n \to \infty}\prod_{k=n}^{\infty}\left(1 - ...
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0answers
23 views

Convergence in probability (in measure) of a weighted sum of random functions

Consider a mesh of points $\pi_{n} = (t_{1n},\ldots,t_{K_{n}{n}})$ with $0 < t_{1n} < \ldots < t_{K_{n}n} < 1$ and weights $(w_{1n},\ldots,w_{K_{n}n})$ such that ...
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1answer
84 views

Find the limit of $\sum\limits_{n=0}^\infty \frac{n}{3^n}$ [duplicate]

Hi all What would the best way/method be to approach this, any advice would be appreciated
1
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2answers
28 views

Find relative radius of convergence for this seies

Given two series $\sum _{n=1}^{ \infty} a_nz^n$ and $\sum _{n=1}^{ \infty} b_nz^n$ who both have radius of convergence $R$, show that the radius of convergence for $\sum _{n=1}^{ \infty} c_nz^n$ is at ...