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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 .

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Show that the function converges at a = 0

Show that the function $ f:(−1,0)∪(0,1) → R , x → { 1 \over {1 \over x} +1} $ converges at $ a = 0$ The function ${ 1 \over {1 \over x} +1}$ is defined on the interval $ f:(−1,0)∪(0,1) $ excluding the point $x = 0$ because the denominator $ {1…
I don't need a name
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Convergence of $\lim_{n \to \infty} \ln(n) \cdot \prod_{\text{p prime} \le n} (1 - \frac{1}p)$

The expression $\lim_{n \to \infty} \ln(n) \cdot \prod_{\text{p prime} \le n} (1 - \frac{1}p)$ seems to converge to 0.5614372$\dots$ Assuming it converges to $\lambda = 0.5614372\dots$ does that imply $\lim_{n \to \infty} \prod_{\text{p prime} \le…
amitbashan
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Sum of decimal values from specific range.

When I was playing maths in python, I found some specific equation. The main idea was to sum up all decimals of values from between a specific range. The range is from $a^2$ to $(a+1)^2$. It's because for this borderline values the $\sqrt{x}$ is a…
Michal
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Definition of Cauchy's Convergence Test using $\lim$ rather than $\limsup$

I am looking at The ''Penguin Dictionary of Mathematics'', editions 2 and 4, edited by David Nelson (1998 and 2008 respectively). I see the entry for "Cauchy convergence test" and I see this: Let $a_1 + a_2 + \cdots + a_n + \cdots$ be an infinite…
Prime Mover
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The convergence or divergence of the series formed by the minimum values of two monotonically decreasing divergent series of positive terms.

Given two sequences $\{a_n \}$ and $\{b_n \}$ that are both monotonically decreasing sequences of positive terms, and the series $\sum a_n$ and $\sum b_n$ both diverge, where $c_n=\min(a_n, b_n)$, can we determine the convergence or divergence of…
linchen
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Unknown limit of convergence

Are there any function or infinite series which is known to converge to a value, but whose value of convergence is unknown (impossible to prove)?
pseudobulbose
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Does series converge or diverge

I need to figure out if the following series converges or diverges: $\sum_{n=1}^{\infty} (\frac{n}{n+2})^{(n(n+1))}$ I tried using Cauchy's criterion: $C_n = (\frac{n}{n+2})^{n+1}$ It seems to be less than 1 so I thought the series should converge…
Alisa Petrova
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Does $\sum_{n=1}^\infty \sin \frac a n$ converge?

Examine for convergence the series $\sum_{n=1}^\infty \sin \frac a n$. The series is divergent if we consider $a_n = \sin \frac a n$, $b_n =\frac{ a}{n}$ then apply the comparison test. But this series is not a series with positive…
Riya Garg
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Contradiction to Bolzano Weierstraß Theorem?

Consider the sequence $(x_n)_{n\in\mathbb{N}} = 2+ \frac{1}{n}$ in $\mathbb{C}$ with the metric $$d(x,y) = \begin{cases} |x-1| + |y-1|&\text{if $x\neq y$}\\ 0 &\text{if $x=y$} \end{cases} $$ Then one is able to show that this is a bounded sequence…
HelloEveryone
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How to find the values of q for which the integral $\int_3^\infty \frac{(1 + x^{25q})(1 - x^q)}{((1 + x^q)(1 - x^{25q})}\, dx$ diverges?

For the integrand $\frac{(1 + x^{25q})(1 - x^q)}{((1 + x^q)(1 - x^{25q})}$, I calculated that the dominant term in the numerator is $x^{26q}$ and the dominant term in the denominator is also $x^{26q}$. So the integrand simplifies to 1. What does…
Crystal
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Subtraction of Big O notation

Imagine that I have: $$\sum_{i = 1}^n O(n^{-1/2}) - \sum_{i = 1}^n O(n^{-1/2}).$$ By rewriting in the following way we could have $$\sum_{i = 1}^n O(n^{-1/2}) - \sum_{i = 1}^n O(n^{-1/2}) = \sum_{i = 1}^n O(1)n^{-1/2} - \sum_{i = 1}^n O(1)n^{-1/2} =…
Eryna
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The following series converge or diverge? Prove it!

Does the following series converge or diverge? $$\sum_{n=1}^{\infty}e^{-n+\sqrt{n}} n^4$$ I tried using the root test to prove convergence. So we take $\quad\lim_{n\rightarrow\infty}\sqrt[n]{a_n}\quad$, but I don't know how to find this limit. I…
222111
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Does $\sqrt[x!]{x}$ converge? If yes, to what?

Does $\sqrt[x!]{x}$ converge? If yes, to what? I couldn't find any solutions to it. If anybody could show to what it converges or diverges, then I would appreciate it!
Csoriburi
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Once you have the interval of convergence, how do you find the values of x for which the series converges absolutely and conditionally?

Take this series: $$\sum_{n=0}^{\infty} \frac{nx^n}{n+9}.$$ I found the interval of convergence to be $-1 < x < 1$ but how do I approach finding the interval for which the series converges absolutely and conditionally? I understand that for it to…
Bill Cogn
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Calculation of speed of convergence

Let consider this expression $\binom{n}{k}p^k(1-p)^{n-k}$ I understand that if $n$ is very large then $(1-p)^{n-k} \to 0$ . Therefore entire expression converges to 0. However how can I show that that the rate of convergence is $O(n^{-1/2})$?
Brian19931
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