Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [convergence-divergence]

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

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
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Why does this integral diverge when $\alpha \gt 2$?

Problem text: For which values of $\alpha$ does the integral below converge? $$\int\int_{D}\frac{1}{(x+y)^{\alpha}}dxdy$$ Where $D: 0 \le y \le 1 - x, 0 \le x \le 1$. Answer: $$\int\int_{D}\frac{1}{(x+y)^{\alpha}}dxdy=\int _0^1\left(\:\int…
Tsiolkovsky
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To what value infinite sum converges to

What is the limit of $$\sum_{n=1}^{m} \frac{n}{1.3.5...(2n+1)}$$ when $m \to \infty$ ? I tried to find a recursive formula but failed to put things on place ? Also I had a little doubt that when we say a series is converging then does we mean that…
RKK
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Is $\sum_{n=3}^\infty \frac{1}{n^2 \log^3 n}$ absolutely convergent?

Is $$\sum_{n=3}^\infty \frac{1}{n^2 \log^3 n}$$ absolutely convergent? Using comparison test, since $n$ is greater than or equal to 3: $$\frac{1}{n^2 \log^3 n} \leq \frac{1}{n^2}$$ And, we know $$\sum_{n=3}^\infty \frac{1}{n^2}$$ converges by…
Risa
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stable, consistent and convergence of one step method

the one step method defined by $x_{k+1} = x_k + hγ(t_k,x_k)$ for the ODE $dx(t)/dt = f(t,x(t))$ with $γ(t_k,x_k) = f(t_k+h/2,x_k+(h/2)*f(t_k,x_k))$. what is the conditions for the convergence of one-step methods considered above And,…
crazy
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Almost sure convergence implying mean square convergence

A sequence $(X_n)_{n\in\mathbb{N}}$ is said to converge to $X$ in the almost sure convergence sense if $\lim_{n\rightarrow\infty} X_n \rightarrow X$ on a set with probability $1$. Then I proceed like below \begin{align}\lim_{n\rightarrow\infty} X_n…
learner
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find speed from object size growth in video

if I have a video of a car coming at me, can I find the speed it is traveling by the growth of the car over time. for example, if it takes two seconds for the height of the car (from my perspective) to double, can I find out how fast it is coming at…
Mark
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To investigate the convergence of integrals

I want to investigate the convergence of integral $$\int_0^1 \frac{1}{\sqrt[3]{- x^{4} + 1}}\, dx$$ How shall I deal with it? I thought about taking $\frac{1}{x^p}$ and finding $p$ by $\lim\limits_{x \to 1} \frac{f(x)}{g(x)}$. Now I decided to use…
TaumCkBep
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Convergence of Type II Improper Integral

State whether the following integral is convergent or divergent:$$\int_{0}^{1} \frac{\sin(x)}{x^{1.5}} \ dx $$ The answer says that it converges due to a comparison with $\frac{1}{\sqrt{x}}$. I don't see how this works, as $\frac{1}{\sqrt{x}}$ is <…
StopReadingThisUsername
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Radius of convergence of $x$

What is the radius of convergence of $f(x)=x$? I have found the taylor series of x to be just = $1 + (x-1)$ , but then how can I use the ratio test to determine the radius? Or is there another way?
L.mak
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Determine convergence $\sum_{n=0}^{\infty}\frac{n!\cdot 2^n}{1\cdot 3\cdot 5\cdot \cdot \cdot (2n+1)} $

I tried Ratio test for this series and got 1. $$\sum_{n=0}^{\infty}\frac{n!\cdot 2^n}{1\cdot 3\cdot 5\cdot \cdot \cdot (2n+1)}$$ Any suggestions? I believe the limit of the sequence goes to $\infty$ , but I don't know how to prove. Thanks a lot!
bffaf02
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Test convergence of the series $\sum(2\sqrt{n+1}-\sqrt{n+2}-\sqrt{n})$

$$\sum(2\sqrt{n+1}-\sqrt{n+2}-\sqrt{n})$$ I have tried comparison test but gives the best I get was $\dfrac{1}{\sqrt{n}}$.
UfmdFkiF
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Is $\sum_{n=1}^\infty \frac{\sqrt{n^3+1}}{n^2}$ convergent?

My first take has been to get rid of the root by doing this: $$\sum_{n=1}^\infty \frac{\sqrt{n^3+1}}{n^2} =\sum_{n=1}^\infty \sqrt{\frac{n^3+1}{n^4}}$$ Therefore, if $\sum_{n=1}^\infty \frac{n^3+1}{n^4}$ is convergent, the original series must be…
user403851
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Comparison test.

I want to show that $\sum_{n=1}^{\infty} \sum_{k=n}^{\infty}\frac{1}{k^{3}} < \infty$. Therefore I want to show that $\sum_{n=1}^{\infty} \sum_{k=n}^{\infty}\frac{1}{k^{3}}$ converges. From Wolfram-Alpha I can conclude this by the comparision…
whoisitnow
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How can we prove that $f(x) = \lfloor 1.5x \rfloor$ is not onto?

How can we prove that $f:\mathbb Z \to \mathbb Z, f(x) = \lfloor 1.5x \rfloor$ is not onto? While it's very obvious to see, how can I actually prove that this is in fact the case? Traditional methods that I would use don't seem applicable because of…
acak55
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Convergence of functions in norm and pointwise

I understand that $L^2$ convergence does not imply pointwise convergence and vice versa. But I think that $L^2$ convergence must imply pointwise convergence a. e. $x$? So, since Schwartz functions are dense in $L^2$, then, for every $f\in L^2$,…
Marina
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