# Tagged Questions

Convergence of sequences and different modes of convergence.

1answer
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### How to determine when a rounded sequence “converges” and what the convergence value is

I have a process on a server that iteratively computes a value over time. The value follows a fairly simple formula, which would generate a convergent sequence; however, the value of the formula is ...
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### How can I show this sequence $u_n$ is divergent: $u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

How can I show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
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### Will the expression $\sum_{i=1}^{n}{\frac{i^{2}}{n^{2}}}$ converge as n approches infinity?

I have the following expression: $$\lim_{n \to\infty}\ \sum_{i=1}^{n}{(\frac{i}{n})^{2}}$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to figure it out?
5answers
64 views

### Does this expression diverge or converge?

I have the following expression: $$\lim_{n \to \infty} \frac{2}{n^2} \ {\sum_{i=1}^{n}{\sqrt{n^2 - i^2}}} \$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to ...
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### Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
3answers
168 views

### Convergence of $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
1answer
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### Can you prove the convergence of $\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx$?

Can you prove the following improper integral is convergent? $$\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx.$$
7answers
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### Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty$. , $n\in \mathbb{N}$ My progress LHS is ...
1answer
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### Rate of convergence vs number of iteration

Can anyone explain to me the difference between rate of convergence and number of iterations for a numerical algorithm? Is it correct to say rate of convergence measure how fast the sequence approach ...
1answer
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### Behavior of fundamental solution to heat equation after projection

I am considering the behavior of $$\frac{1}{h}\|(1-P_h)S(h)v\|,\tag{1}$$ and $$\frac{1}{h}\|(1-P_h)S(h)P_hv\|,\tag{2}$$ as $h\to 0^+$ for a fixed good enough $v$. I hope to show one of them converges ...
0answers
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### Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
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### How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text.$$ I tried to apply Chebyshev's inequality, ...
17answers
14k views

### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
1answer
100 views