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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Understand Almost sure convergence in a paper

I am reading a paper and to strengthen my understanding I would like to know why $Z_{n} = \sum_{k=0}^{n} \frac{\Delta_{k}}{P(N\geq k)}\mathbf{1}_{N\geq k}$ converges almost surely to $Z= \sum_{k=0}^{N}\frac{\Delta_k}{P(N\geq k)}$, where $\Delta_{n}…
Maths student G
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Let $nx_n \to a \in \mathbb{R}\setminus\{-\infty, 0, \infty\}$. How to find $\lim\limits_{n\to\infty}\sum_{k=1}^{n}{x_k^2}$

Let $nx_n \to a \in \mathbb{R}\setminus\{-\infty, 0, \infty\}$. How to find $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}{x_k^2}$ For example for $x_n=\frac{a}{n}$ we have that…
Auguste Ladislao
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does the series $ \sum _ { n = 0 } ^ \infty \frac 1 { \sqrt n + \ln n } $ converge or diverge? having trouble

Does the series $$ \sum _ { n = 0 } ^ \infty \frac 1 { \sqrt n + \ln n } $$ converge or diverge? I believe this is divergent but having a hard time proving it is.
user851884
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Reformulating an expression to use Gauss's test

I have seen the following two cases for convergence testing using Gauss's test- $\frac{u_j}{u_{j+1}} = \frac{(2j+1)(2j+2)}{2j(2j+1)-\lambda}$ For large j, my textbook reformulates the RHS expression to- $\frac{2j+2}{2j} + \frac{B(j)}{j^2} = 1 +…
Paddy
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Showing convergence in probability for poisson distribution

Given random variable X and N so that, N ∼ Poisson(λ), and X|N ∼ Bin(N,p) where p is a constant (Assume that X = 0 when N = 0 and 0 < p < 1). Note that the moment generating function of a Bernoulli random variable with parameter p is 1 − p + etp,…
Greyson21
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Two definitions of order convergence

I came accross two different definitions of order convergence, one (Def 1) given by my professor (I can't find any same ones elsewhere), and one (Def 2) which can be found in any materials available on the Internet, such as here. $\{p_k\}$ is an…
user36706
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Convergence of definite integrals given unknowns.

Given the following equation, $$ \int_{1}^{2} \frac{1}{x(\ln x)^{a}} dx $$ I am supposed to find the condition on the constant $a$ such that the above integral is convergent. Substituting $u = \ln(x)$, we get $$ \frac{du}{dx} = \frac{1}{x} $$ Where…
Naja
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Calculate the modulus of uniform continuity of the functions in $R$,

Calculate the modulus of uniform continuity of the functions in $R$, a) $x→ sin (\frac{1}{x})$ para $ x>0$, b)$ x→sin (x^2)$, c) $x→x^2$ Where the modulus of uniform continuity of a function $ f:A→R $ es: $φf(δ):=sup{|f(x)−f(y)|:x,y∈A,|x−y|≤δ}$ for…
taniadiaz
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Convergence of series: three questions and "how do we know when we have the right answer"?

I would like to know if my reasoning for the following questions is correct. In each case determine whether or not the series converges $$ 1. \quad \underset{k=1}{\overset{\infty}{\sum}} (-1)^k \cos\left(\frac{1}{k}\right)$$ $$ 2. \quad…
user9750060
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Convergence of a Sequence $x_{n+1}=\frac{1}{2}(x_n+\frac{3}{x_n})$

Prove the sequence given by $x_1= 1$ and $x_{n+1}= \frac{1}{2}(x_n+\frac{3}{x_n}$) for $n \geq$ 1 is convergent to $\sqrt3$? Struggling to come up with any real idea of where to begin.
Nuemann12
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Do $\sum _{n=1}^{\infty }\:\left(\frac{1}{n\:}\right)^{\ln\:n}$ and diverge?

So I want to find out if $\sum _{n=1}^{\infty }\:\left(\frac{1}{n}\right)^{\ln n}$ and $\sum _{n=1}^{\infty }\:\frac{\left(-1\right)^n}{\ln\left(2+\ln n\right)}$ diverges or not. For $\sum _{n=1}^{\infty }\:\left(\frac{1}{n }\right)^{\ln n}$, I see…
PythonDaniel
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Does the generalised integral $\int_{0}^{\pi}\frac{\sqrt x}{\sin x}dx$ converge or diverge?

Does the generalised integral $\int_{0}^{\pi}\frac{\sqrt x}{\sin x}dx$ converge or diverge? The first thing I would do here is split it into two integrals $$\int_0^\pi \frac{\sqrt x}{\sin{x}}dx=\int_0^{\frac{\pi}{2}} \frac{\sqrt…
PythonDaniel
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conditionally convergence of alternating series

Finding whether the series $\displaystyle \sum^{\infty}_{k=2}\frac{(-1)^k 4^{k}}{k^{10}}$ is absolutely convergent, conditionally convergent or divergent What i try :: For absolutely convergent/Divergent. Let $\displaystyle…
jacky
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How do i show that $x^n$ converges on $|x|=1$ only if $x=1$?

The radius of convergence of $x^n$ is $1$. So the convergence of it is inconclusive when $|x|=1$, but how do i prove that $x$ is divergent when $|x|=1$ but $x\neq 1$, not using gamma function?
Jj-
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limit of a sequence $\frac{(-1)^n}{n}$

Finding Convergence or divergence of sequence $$a_{n}=\frac{2+(-1)^n}{n}$$ What I try :: A sequence $\{a_{n}\}$ is convergent if $\displaystyle \lim_{n\rightarrow \infty}a_{n}=0.$ Otherwise it is diverges. $$\lim_{n\rightarrow…
jacky
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