Convergence of sequences and different modes of convergence.

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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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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
59 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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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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58 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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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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35 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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4answers
107 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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75 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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52 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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64 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
118 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
40 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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38 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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61 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
125 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
62 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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56 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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156 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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60 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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22 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
44 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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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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78 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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52 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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32 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
42 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
27 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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47 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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133 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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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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65 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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161 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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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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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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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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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 ...
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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 ...
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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 ...
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1answer
20 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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36 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 ...
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Power series function - convergence interval

Could someone help me finding the function and convergence interval for following power series? I don't need a step by step answer, but I'm not entirely sure where to start. $\sum_{n=0}^{+\infty} ...
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7 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
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1answer
26 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
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1answer
29 views

Show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$.

This should be a relatively easy question, but I can't seem to figure it out. I want to show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$. As ...
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34 views

Asymptotic Equivalence implies same asymptotic distribution?

A book I'm reading stated that if we have nonnegative random variables, and if $X_n\to X > 0$ in distribution and $\frac{Y_n}{X_n} \to 1$ in probability then $Y_n \to X$ in distribution. However, ...
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1answer
23 views

Convergence of expecations implies convergence of positive and negative parts?

If we have $E|X_n| \rightarrow E|X|$ does that imply \begin{equation} \lim_{n\rightarrow\infty} E X_n^\pm = X^\pm \end{equation} How about if we only have $EX_n \rightarrow EX$? Is this true in ...
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26 views

Convergenge of the error in Poisson

I have this open exercise which is difficult for me to answer completely I know that h - density of the mesh C - shape of the area, no load function Factors affecting the left-hand side of the ...
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1answer
37 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...