Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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2answers
19 views

Convergence and calculation of a particular series.

Let $$p_n(x)=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+\frac{x^4}{(x+1)(x^2+1)(x^4+1)}+...…+\frac{x^{2^n}}{(x+1)(x^2+1)(x^4+1)(……)(x^{2^n}+1)}$$ I know this can be simplified to $$p_n(x)=1-\frac{1}{(x+1)(...
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3answers
49 views

Let $a_{n} = \sqrt{n^{2}+n} - n$, for $n\in\textbf{N}$. Is the sequence $(a_{n})_{n=1}^{\infty}$ monotonic?

Let $a_{n} = \sqrt{n^{2}+n} - n$, for $n\in\textbf{N}$. Show that $a_{n}$ converges as $n\to\infty$. What is the limit? Is the sequence $(a_{n})_{n=1}^{\infty}$ monotonic? MY ATTEMPT The answer to the ...
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0answers
66 views

Does $\iiint_{\mathbb{R}^3} \frac{1}{1+x^2y^2z^2}$ converge?

Does $\iiint_{\mathbb{R}^3} \frac{1}{1+x^2y^2z^2}$ converge? My try from symmetry it's enought to look when $x,y,z\geq 0$ and changing variables to $u=xyz,v=y,t=z$ $\frac{\partial (u,v,t)}{\partial(x,...
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1answer
17 views

If $X_n$ converges to $X$ in $L_p$ and $Y_n$ converges to $Y$ in $L_p$ then $X_n + Y_n $ converges to $X + Y$ in $L_p$ [closed]

I want to show that if $X_n \xrightarrow{L^p} X$ and $Y_n \xrightarrow{L^p} Y$ then $X_n + Y_n \xrightarrow{L^p} X + Y$ ($p \geq 1)$. My idea is to use the following facts (whose proofs I won't give ...
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1answer
28 views

Find the solution to these series

Find the accumulation points or the limit if the sequence converge $a_n=\frac{4\sqrt[3]{5n^7-3n^2}}{2\sqrt{n}+3n^2}$ $a_n = \frac{e^{in}}{n}-e^\frac{-in}{n^2+1}$ $a_n = e^{(1+\frac{(-1)^n}{n})^n}$ ...
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1answer
30 views

Convergence in probability implies mean squared convergence

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{F}$ measurable random variables. Let $X$ be another $\mathcal{F}$ measurable ...
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2answers
22 views

Show that convergence in probabiltiy plus domination implies $L_p$ convergence

I want to show that if random variable $X_n $ converges to $X$ in probability (Let $(\Omega, \mathcal{A},P)$ be the probability triple) and $|X_n| < Y \,\,\forall\, n$ then $X_n$ converges to $X$ ...
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Determine if $\int_1^{\infty}\frac{dx}{x^p+x^q}$ converges …

Determine if $\int_1^{\infty}\frac{dx}{x^p+x^q}$ converges if $\min(p, q) < 1$ and $\max(p, q) > 1$, where $\min (p, q)$ is the minor of the numbers $p$ and $q$, and $\max (p,q)$ is the major ...
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convergence or divergence of infinite rational series

Finding whether the series $$\sum^{\infty}_{k=0}\frac{5k^2+7}{8k^2+2}$$ is converges or diverges. What i Try: I am Trying to solve it using ratio test Let $\displaystyle a_{k}=\frac{5k^2+7}{8k^2+2}$. ...
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1answer
90 views

Does the following series converge or diverge: $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$?

Which of the following series converge, and which diverge? $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$ $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$ $\displaystyle\sum_{n=1}^{\infty}\...
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Show that $x=\lim_{n\rightarrow \infty} x_{n} $ exists, where $x_{n+1}=\frac{1}{2}(x_{n}+\frac{4}{x_n})$.

Let $x_0>0$, and $x_{n+1}=\frac{1}{2}(x_{n}+\frac{4}{x_n})$. Show that $x=\lim_{n\rightarrow \infty} x_{n} $ exists. My attempt: Let $x_0 \geq 2$, and $x_{n+1}=\frac{1}{2}(x_{n}+\frac{4}{x_n})$. $\...
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27 views

Radius of convergence of a power series: seemingly two different approaches

From a textbook, I have learned that the radius of convergence of a power series, i.e., $\sum_{n = 0}^{\infty} c_n (x - a)^n$, can be obtained from the limit (if it exists): $\lim_{n \to \infty} |\...
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1answer
24 views

What is the convergence radius of 1/(sinx)^2? [closed]

What is the converges radius of 1/(sinx)^2 if we expand it over 2?
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30 views

Is $\sum_{n=1}^{\infty} \frac{a_n}{(\sum_{k=1}^n a_k)^2}$ convergent whenever $\sum_{n=1}^{\infty} a_n =\infty$ for unbounded $a_n >0$?

Let $a_n>0$ be an unbounded sequence. Suppose $\sum_{n=1}^{\infty} a_n =\infty$. I want to show that $$\sum_{n=1}^{\infty} \frac{a_n}{(\sum_{k=1}^n a_k)^2}$$ is convergent. What I have tried: We ...
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1answer
31 views

The condition and proof about the integral test for convergence

The proof about the integral test: Suppose $f (x) $ is nonnegative monotone decreasing over $[1,\infty)$, then the positive series $\sum_{n=1}^{\infty}f(n)$ is convergent if and only if $\lim_{A\...
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Decide on the convergence of the series [closed]

Decide on the convergence of the series: $$\sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n^{p+{1 \over n}}} $$ , p ∈ R
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Why can the $n_{\epsilon}$ of the definitions of convergence and Cauchy sequence be the same in the following proposition?

I have the following proposition proved in my lectures notes, but I think there are a couple of errors and there is one think I don't get: If $p_n$ is a Cauchy sequence in a metric space $(X,d)$, the ...
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3answers
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Let $x_n\to a, y_n\to a$, $\lim_{n\to\infty}\frac{x_n}{y_n}=1$, $f$ continuous, $f(a)=0$. Show that $\lim_{n\to\infty}\frac{f(x_n)}{f(y_n)} =1$. [closed]

Does this generally hold? Does it require anything beyond continuity of f? In a metric space $S\subset \mathbb{R}^n$, continuity of $f$ is equivalent to sequential continuity, meaning that $f$ is ...
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4answers
59 views

Prove monotonicity of a sequence. [duplicate]

$1)$ I have a sequence defined in terms of a recurring relationship. Namely, $$s_0=\sqrt{2},\;s_{n+1}=\sqrt{2+\sqrt{s_n}}.$$ I could prove that it is bounded, but having difficulty in proving that it ...
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21 views

Q-linear convergence rate

I have a question regarding the Q-linear convergence of sequences. I know that the definition is: $\exists \delta \in (0,1)$ such that $$ \|x^{k+1}-x^\star\| \leq \delta \|x^{k}-x^\star\| ~(1)$$ In ...
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1answer
39 views

What is the convergence radius of $ f(z) = \dfrac{z+2}{(z-i)^3 (z-5)^2}$? [closed]

Consider the series expansion of the complex function $f(z) =\cfrac{z+2}{(z-i)^3 (z-5)^2}$, centered in $(1, 0)$. How do I find its radius of convergence?
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1answer
36 views

The series $ Σ=(-1)^n 2^n z^n $ from n=0 to infinity wtih |z| <1/2 where it converges? [closed]

Where does this series converges $ Σ=(-1)^n 2^n z^n $ with |z|<1/2 I am having problem to find solution on this one. How would you aproach it ? EDIT: i dont know how to solve when i know that |z|&...
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$\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius [closed]

Proof: Suppose power series $\sum_{n=0}^{\infty}a_n x^n $, if series ${a_n}$ is divergent. Then $\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius. ...
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1answer
117 views

Prove $\int_{0}^{+\infty}\frac{1}{f(x)}dx$ is convergent when $\int_{0}^{+\infty}\frac{e^x}{(e^xf(x))'}dx$ is convergent

Suppose $f(x)$ is positive monotone increasing function over $[0,\infty)$, and it has derivative. Prove: if $\int_{0}^{+\infty}\frac{e^x}{(e^xf(x))'}dx$ is convergent, then $\int_{0}^{+\infty}\frac{1}{...
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1answer
64 views

If $\sum a_n^k$ converges for all $k \geq 1$, does $\prod (1 + a_n)$ converge?

By definition, an infinite product $\prod (1 + a_n)$ converges iff the sum $\sum \log(1 + a_n)$ converges, enabling us to use various convergence tests for infinite sums, and the Taylor expansion $$ \...
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1answer
50 views

Prove that if there exists a subsequence of $a_{n}$ which converges to $L$ , then $L$ is a limit point of $a_{n}$.

Let $(a_{n})_{n=0}^{\infty}$ be a sequence of real numbers, and let $L$ be a real number. Then the following statements are logically equivalent: (a) $L$ is a limit point of $(a_{n})_{n=0}^{\infty}$. (...
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2answers
52 views

Showing a series converges absolutely

The goal is to prove that if $|\frac{c_{n+1}}{c_n}|\leq1+\frac{a}{n}$, where $a<-1$ and $a$ does not depend on $n$, then the series $\sum_{n=1}^\infty c_n$ converges absolutely. My idea: to have ...
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0answers
14 views

What is the theory of the cones of minorant functions?

As a non - mathematician, I face some difficulty in understanding concepts such as convex cone, minorant/majorant function. The question is What is the theory of the cones of minorant functions? And ...
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24 views

How can i find the sum of the infinite series? [duplicate]

I am having trouble finding the sum of this series. I was able to show that it is convergent by Ratio Test but i could not find the sum of this series. I would appreciate it very much if you could ...
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0answers
56 views

Convergence of an improper triple integral

Let $$V=\{(x,y,z) \in \mathbb{R^3} : x\ge 1,y>0, x^2+z^2<\mathrm{min}(2x,y^2)\}$$ be a region of the real space. For which values of $\alpha \in \mathbb{R},$ $\iiint_V |z|y^{-\alpha}$ converges (...
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3answers
29 views

Series convergence test, $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$

I'm trying to find all $x$ for which $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$ converges. I know I need to check the ends ($-1$ and $5$) but I'm not sure what to happen after that. I'm pretty sure I'...
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1answer
59 views

Prove the uniform convergence of ${\frac{1}{(1+ {\frac{y^2}{n}})^n}}$

How can I prove the uniform convergence for these two tasks: $${\frac{1}{(1+ {\frac{y^2}{n}})^n}} ⇉ e^{-y^2} $$ I understand that the function on the right is the limit function, because: $$\lim _{{...
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0answers
36 views

Limit and expectation of bounded random variables

Let $X_n$ and $Y_n$ be two stochastic processes with $X_n \leq 1$ and $Y_n \leq 1$ for all $n$ (not independent). Further, suppose $$ \frac{E (X_n)}{q^n} \to 1 \quad \text{and} \quad Y_n \to 1 \text{ ...
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1answer
65 views

What's this partial sum $ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

I want to get this partial sum of $$ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$$ which it is convergent and it is closed to one half , I have tried to use polylogarithm function which is defined as :...
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1answer
41 views

Question on a problem regarding improper integral

I'm studying improper integrals with Paul's Online Notes as a reference. Sorry if I'm quoting it here, but the website has the following problem: Determine if the following integral is convergent or ...
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2answers
52 views

The antiderivative of $\sum_{n\gt 0}\frac{x}{n(x+n)}$

I tried to calculate $\int\sum_{n\gt 0}\frac{x}{n(x+n)}\, \mathrm dx$: $$\begin{align}\int\sum_{n\gt 0}\frac{x}{n(x+n)}&=\sum_{n\gt 0}\frac{1}{n}\int\left(1-\frac{n}{x+n}\right)\, \mathrm dx \\&...
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34 views

Determine whether the series converges or not: $\sum_{j=1}^{\infty} \left(e^{(-1)^{j}\sin(1/j)}-1\right)$

Determine whether the series converges or not: $$ \sum_{j=1}^{\infty} \left( e^{(-1)^{j}\sin(1/j)} - 1 \right) $$ My attempt: \begin{align*} &\sum_{j=1}^{\infty} \left( e^{(-1)^{j}\sin(1/j)} - 1 \...
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0answers
43 views

Convergence of series over full modular group

I have a question regarding the series over the full modular group $\Gamma=\{\bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)\mid a,b,c,d\in\Bbb Z, ad-bc=1\}$ $$R_n(z):=\sum_{\...
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1answer
52 views

Is $\sum_{n=1}^{\infty}$ $(\sqrt[\leftroot{-3}\uproot{3}n]{a}-1)^p$ divergent or convergent?

I know $a>1,p>0$. I've tried the direct comparison test, but I can´t find a sequence $b_{n} > a_{n}$ whose sum converges, or a sequence $b_{n} < a_{n}$ whose sum diverges.
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1answer
34 views

Probability convergence and almost surely convergence [closed]

Let $(\Omega, \Im ,\mathbb{P})$ be a probability space and let $\left\lbrace X_{n} \right\rbrace {}_{n}$ be a sequence of random variables such that $$\mathbb{P} \left(X1>X2>...>0\right)=1$$ ...
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0answers
42 views

Lipschitz Constant for Equation System

I'm would like to compute the region of convergence from Newton-Rapson Method for a multivariate system of equations by a theorem that states: Let $\vec{F}: \mathbb{R}^N \rightarrow \mathbb{R}^N$ be ...
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2answers
41 views

Abbott's proof that any rearrangement of an absolutely convergent series converges to the same limit as the original

Here is his proof in full: Assume $\sum\limits_{k = 1}^{\infty} a_k$ converges absolutely to $A$, and let $\sum\limits_{k = 1}^{\infty} b_k$ be a rearrangement of $\sum\limits_{k = 1}^{\infty} a_k$. ...
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2answers
39 views

How to find domain of convergence and sum for such series: $\Sigma^{\infty}_{n=1}\frac{x^{2n}}{1+x^{4n}} $

$$\Sigma^{\infty}_{n=1}\frac{x^{2n}}{1+x^{4n}} $$ Given that $1+x^{4n}$>x^{4n} take a series $\sum^{\infty}_{n=1} \frac{x^{2n}}{x^{4n}} = \sum^{\infty}_{n=1} \frac{1}{x^{2n}}$, then for $|x| > ...
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1answer
45 views

Find the sum for the series $\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x}$

So here is the series: $\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x}$ . Because $\cos (\pi n) = (-1)^n$ I can rewrite the series as: $\sum^{\infty}_{n=1}\frac{(-1)^...
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1answer
26 views

Question regarding the correlation of limit points and convergent sequences

I've been told, that: $(i)$ $M$ is a closed set $(ii)$ For every convergent sequence $a_{n}$ in $M$, $\lim \limits_{n \to \infty } a_{n} \in M$ are equivalent. I understand that $M$ being a closed ...
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1answer
49 views

Find the domain of convergence for the series as well as the sum $S(x)$.

The given series: $$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x}$$ Here is what I did: $$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \...
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1answer
70 views

Whether $\lim_{n\to \infty} \frac{2}{\mathsf{e}}(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k}(1-\frac{2k}{n})^{n-1})^{-1/n}$ exists

Problem: Decide whether or not $\lim_{n\to \infty} \frac{2}{\mathsf{e}}\left(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k} \left(1-\frac{2k}{n}\right)^{n-1}\right)^{-1/n}$ exists. Background ...
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1answer
25 views

Convergence Theorem for Power series : radius of convergence and normal convergence

I am currently working through the textbook "Complex Analysis" by Freitag and Busam. Proposition III.2.1 (Convergence Theorem for Power Series) reads : For each power series $$ a_{0} + a_{1} ...
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1answer
24 views

How do I prove that a sequence of functions is convergent?

For $ n \in \mathbb N$ $f_n (x) = \frac{n^2 }{(x^2 +n^2)}$ Let $x \in \mathbb R $. Prove that $f_n (x) \rightarrow 1$ as $n \rightarrow \infty$ My attempt: Let $\epsilon>0 $, then by definition of ...
2
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2answers
54 views

Complicated Improper Integral convergence/divergence

Does the following integral converge for $x < 0$ $$\int _x^0\:\cfrac{\ln^2 \ (|t |)}{ \sqrt[3]{t} }dt$$ I tried splitting it into two separate integrals under the assumption that both are ...

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