Convergence of sequences and different modes of convergence.

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1answer
15 views

How to test convergence for a tetration series slightly below the harmonic series?

I have the following series to test convergence \begin{align} S_{\infty}= \sum_{n=1}^{\infty} \dfrac{1}{n} \left( \dfrac{1}{n} \right)^{ \left( \dfrac{1}{n} \right) } < \sum_{n=1}^{\infty} ...
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2answers
32 views

Sequence Convergence using bounding sequences

Can someone help me? The hint they give me is to find two bounding sequence, but I don't understand how this could help me Thanks
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1answer
13 views

Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
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2answers
135 views

Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: ...
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1answer
55 views

A sequence of nonconstant i.i.d. random variables converges with probability zero

Proove: $X_{n} iid, X_{n}$ not constant a.s. $\iff P(X_{n}$ $converges)=0$ My idea for "$\Rightarrow$": $X_{n}$ not constant a.s. $\iff \forall$ c $\in \mathbb{R}$, $\varepsilon$ > 0: ...
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3answers
44 views

Convergence of series $\sum$$u_n$= $\sum$$\frac{n! x^n}{(n+1)^n}$

My series is $$1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots$$ My approach: $$u_n= \frac{n! x^n}{(n+1)^n}$$ So, $$u_{n+1}= \frac{(n+1)! x^{n+1}}{(n+2)^{n+1}}$$ So, ...
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0answers
24 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
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1answer
50 views

Test for convergence of the series.

$$\frac{1}{1 \cdot 3}\ + \frac{2}{3 \cdot 5}\ + \frac{3}{5 \cdot 7}\ + \cdots$$ What is the $nth$ term here and what test should I use?
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34 views

Determining if series converges or diverges

The Series is For this series the ratio test is inconclusive. I have rewritten the series as Currently i am approaching the problem using limit test. I couldn't progress from this point. Any ...
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3answers
43 views

Is this positive term series convergent?

My series is: $\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$ I see my $nth$ term is $\frac{1}{1+2^{-n}}$ How do I test for its convergence?
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1answer
23 views

Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?
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1answer
54 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
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1answer
41 views

Let m ∈ N. Define the relation ≡^ on Z by a ≡^ b for a, b ∈ Z if and only if a ≡ ±b (mod m).

(In other words, the relation ≡^ holds if either a ≡ b (mod m) or a ≡ −b (mod m).) Prove that the relation ≡^ on Z is transitive. ======= I believe there are 3 properties that it must meet ...
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1answer
34 views

Is my proof ok? Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive.

Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive. If $A$ is congruent to $B$ mod $m$ then $A - B = k m~~$ (1) If $B$ is congruent to $C$ mod ...
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45 views

Convergence of averaged sine function

I have stumbled upon those two problems which I got a little stuck on that is show convergence or divergence for the series $$\sum_{n=1}^{+\infty}\frac{\cos(n)}{n}$$ and ...
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1answer
74 views

How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$ \frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-..... $$ Given: $0<x<1$ I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$ My approach was to use Dirichlet's test. ...
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1answer
50 views

Prove that the series is convergent: 1- (1+1/3)/2 + (1+1/3+1/5)/3-…

I can see that this is an alternating series with the $n$-th term $$(-1)^{n+1}\frac{1+\frac13+\frac15+\cdots+ \frac{1}{2n-1}}{n}.$$ What test can I apply to show that it converges? Also, it ...
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0answers
58 views

Law of large numbers weak vs strong

Does someone have an example where the strong law of large numbers do not hold, but the weak law do hold ? If you think there is no such example, please explain why there are 2 laws of large numbers ...
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4answers
115 views

Find the sum $\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots\infty$

How do I find the sum of the following infinite series: $$\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots\infty$$ The series ...
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1answer
38 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
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1answer
36 views

Limit of a (pseudo) non-increasing sequence

Consider a non-negative sequence $t_1,t_2,...$ that is also bounded above? Suppose that the sequence is "pseudo non-increasing" in the sense that $t_{n+1} \leq t_n + e_n$ where $e_1 + e_2 + ...$ is ...
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3answers
54 views

Prove convergence of geometric sequence without Bernoulli's inequality?

Is there an elegant way of proving the convergence of $|q|^n$ for $|q| < 1$ or the divergence of $|q|^n$ for $|q| > 1$ that does not use $(1+x)^n \geq 1+nx$ for $x \in [-1; \infty)$, $n\in ...
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4answers
99 views

Convergence in measure implies pointwise convergence?

In showing that we can replace pointwise convergence with convergence in measure in the Lebesgue Dominated Convergence Theorem, I made the following claim: 1.) $f_n\to f$ in measure ...
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1answer
59 views

If a sequence of random variables converges to both $X$ and $Y$ almost surely, then $X$ and $Y$ have the same distribution

Show that if .${X_n}\mathop \to \limits^{a.s} {\rm{ }}X$. and ${X_n}\mathop \to \limits^{a.s} {\rm{ }}Y$ ,then X and Y have the same distribution. Proof Let $A = \{ \omega \in \Omega :X(\omega ) ...
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1answer
53 views

Convergence of sequences such as $ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$

Examine the following arithmetic sequences if they converge or do not.The first one is $$ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$$ and the second $$C(n)=n/(n^2+1)+\dots+n/(n^2+n)$$ It was on our ...
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143 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
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4answers
57 views

How do I prove convergence of this recursive sequence, what's the limit? [closed]

I have this sequence: $c_n = c_{n-1} + \frac{0.01}{n}, \ \ c_1 = 0.01$ How do I prove the convergence of this, and what is the limit? EDIT: I was trying to solve the problem of a snail crawling on ...
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58 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
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21 views

Convergence of $\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$

For which $\alpha$ (depending on $n$) does $$\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$$ converge? Examples: For $n=1$ the series turns out to be $$\sum_{i=1}^\infty ...
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1answer
42 views

Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n)) $$ .For the 3th one ...
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1answer
49 views

Absolute convergence of series $\sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1}$

$$ \begin{align} \sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1} \end{align} $$ Determine the values of $z,z\in\mathbb{C}$ so that the series converges absolutely I know that the series converges for ...
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1answer
77 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ [closed]

I need to find out whether this series converges or diverges. $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ Can someone help how to solve it?
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42 views

Definition of Global Convergence

I am confused by the notion of "global convergence" as used in numerical optimization literature, and did not find an exact definition for that yet. Now I try to double check my understanding here. ...
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31 views

Convergence test of certain series

I need to find out whether this sequence converges or diverges using limit comparison test. $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$ I've tried it with the use of sequence ...
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1answer
35 views

Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
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1answer
23 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
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39 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
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3answers
132 views

Approximate summation of the given equation

I have been trying from an hour to approximate the value of $M$ in the equation given below. $$ M = \sum\limits_{i=1}^n\left(\sum\limits_{j=1}^n\left(\sqrt{ i^2 + j^2 }\right)\right) $$ One thing I ...
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1answer
18 views

Convergence of a Sequence of Projection Matrices

Suppose I have a sequence of growing matrices $A_n$, and $B_n$, both of the same size, and both rows and columns are growing at the same rate for each step $n$. Furthermore, we assume that there ...
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1answer
64 views

Weak convergence of scaled elements implies norm convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
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4answers
159 views

Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
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15 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
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1answer
32 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
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3answers
88 views

Convergence or divergence of the integral $\int_0^1 dx/\sin x $

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
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1answer
45 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
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32 views

Asymptotic behaviour / Convergence

Let $0<\omega<\infty, \mu >0$ and $z \in \mathbb{R}.$ In my book, it is written that we have the following asymptotic behaviour: i) Claim: $$\lim_{t \rightarrow \infty} \frac{z ...
5
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0answers
112 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
1
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1answer
21 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
1
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1answer
19 views

Continuity of a map from the 2-plane.

Let $f: \mathbb{R}^{2} \rightarrow X$ be a map where $X$ is a Hausdorff topological space. Assume that the restriction of $f$ on $\mathbb{R}^{2}-\{0\}$ is continuous, and the restriction of $f$ on any ...
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1answer
34 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...