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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Simple function converging to a smooth function almost everywhere

I'm having troubles proving following statement $$ \lim_{\varepsilon \to 0^+} \sum_{i=1}^{N(\varepsilon)} f(\xi_i) \chi_{(x_{i-1},x_i)}(x) = f(x) \qquad \mathrm{a.e. on}\ [0,1]. $$ $f(x)$ is a smooth function with compact support on $[0,1]$,…
goofy
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Is there a formal definition for 'digits per term'?

I've seen the phrase 'digits per term', mostly with regards to algorithms that produce $\pi$. I've seen it here ("Yes, Chudnovsky's formula converges at a steady 14.18 digits per term."), on numerous formulas for $\pi$ ("This gives 50 digits per…
Status
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2 answers

Sums of reciprocals

Suppose $n\ge2$ . Prove that neither the sum $\sum_{i=2}^n \frac{1}{i}$, nor the sum $\sum_{i=2}^n \frac{1}{2i+1}$ is an integer The approach a tried to take, was to show that the denominator is never a divisor of the numerator, but I can't figure…
user61067
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convergence rate for expected value of sample mean

Let $X_1,...,X_n$ be a sample of i.i.d. random variables with mean 0 and finite variance, and let $\bar X$ denote the sample mean. How to show $E(|\bar X|)=O(1/\sqrt{n})$? Thanks.
Justin
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Convergence test of $\sum_{n=2}^{\infty} \frac {1}{n} (\frac {1}{\ln n})^{\frac {3}{2}} $

I need to test for convergence $\sum_{n=2}^{\infty} \frac {1}{n} (\frac {1}{\ln\ n})^{\frac {3}{2}} $. I figured that the comparison test will probably be the best option but I have no idea what to compare it with. Thank you for your help.
Tomáš Svoboda
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Uniform convergence of $f_n(x)=\frac{nx}{(n^2x^2+1)^2}$

Given the function $f_n:[0,1] \to R$ $$f_n(x)=\frac{nx}{(n^2x^2+1)^2}$$ I can show the pointwise convergence $\forall x \in [0,1]: \lim_{n \to \infty} f_n(x) = f(x) = 0$. To show uniform convergence seems to be much harder to me. I show you what I…
leo
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1 answer

convergence of $\sum_{n=1}^\infty\frac{1}{x^{\ln n}}$.

I have the series $$\sum_{n=1}^\infty\dfrac{1}{x^{\ln n}}$$ and I wanna know for what x this series is converge. I think with condensation test if $$\sum_{n=1}^\infty2^n\dfrac{1}{x^{n\ln2}}=\sum_{n=1}^\infty(\dfrac{2}{x^{\ln2}})^n$$ be converged,…
user519686
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1 answer

Is there any version of CLT that does not need independent $X_i$'s?

Is there any version of CLT does not need independent $X_i$'s? The reason I'm asking this is because I'm woking on a moving average problem. I have 3 pieces each by CLT converges in distribution to standard normal. However, when I combine the three…
user52144
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1 answer

Does bounded $\{ \frac{1}{n}\sum_{k=1}^{n}a_k \}_n$ imply $\frac{a_n}{n} \rightarrow 0$?

Let $\{ a_n\}_n$ be a positive sequence. If the sequence $p_n = \frac{1}{n}\sum_{k=1}^{n}a_k$ is bounded, then does this imply $\limsup_{n \rightarrow \infty} \frac{a_n}{n} = 0$?
Rajat
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2 answers

Third order convergence of an iteration scheme

Consider the iteration scheme $x_{n+1}=\alpha x_n(3-\frac{x_n^2}{a})+\beta x_n(1+\frac{a}{x_n^2})$ For third order convergence to $\sqrt 2$, the values of $\alpha$ and $\beta$ are ...... I tried it by plugging $x_{n+1}=x_n=\sqrt a$ as…
Nitin Uniyal
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Convergence of a sequence of a continuously differentiable function

Let $f(x): \mathbb{C}^M \rightarrow R$ be a continuously differentiable function of complex vector $x$. Besides, let us assume that $\{x_n\}$ is a sequence of $x$. If we know that sequence $\{f(x_n)\}$ converges to a value (less than $\infty$), can…
Shahram Shahsavari
  • 103
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1 answer

testing the convergence of complex series and my argument.

I'd like to know whether the complex series $$ \sum_{n=1}^{\infty} \left( \frac{\log n}{n} + i^n \left( \frac{\log n}{n} \right) \right) $$ is convergent or not. I guess it is divergent, because in order for complex series $c_n=a_n+i(b_n)$ to be…
glimpser
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How can I determine the convergence of the following series

So I've been solving convergence tests of series all day and I got stuck on the following three: $\sum_{k=0}^\infty \frac{4+|Cos{k}|}{k^3}$ $\sum_{k=0}^\infty \frac{k!}{k^k}$ $\sum_{k=0}^\infty \frac{1}{4+2^{-k}}$ For the first one I've tried to…
David Mason
  • 103
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2 answers

Find the convergence space of $\sum_{n=1}^\infty \frac{n(x-1)^n}{3^ n (2n-1)}$

Find out the convergence space of the following series $$\sum_{n=1}^\infty \frac{n(x-1)^n}{3^ n (2n-1)}$$ I have found that the convergece radius is equal to $R = 3 $ but i am not sure how to check if it is convergent the closed space $[-3,3]$ or…
K Soe
  • 462
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2 answers

Is $\sum_{n=2}^{\infty} \frac{1}{n\log^2 {n}}$ convergent or divergent?

Cauchy ratio test yields 1 (so it's inconclusive). I have tried this: $$\frac{1}{n \log^2n}=\frac{1}{n \log n \log n}=\frac{1}{\log n^n \log n}\geq \frac{1}{\log n^n -n} \approx \frac{1}{\log n!} $$ Now, since $\sum 1/\log n!$ diverges, the original…
user403851
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