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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 .

21990 questions
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convergence of sequences $|a_{n}-L|<1/n$

$a_{n}=\frac{5n}{4n-3}$ , $L=5/4$ show that for all $n>12$ , $|a_{n}-L|<1/n$. right now, I have: $a_{n}-L=\frac{15}{4(4n-3)}<\frac{4}{4n-3}$ since $15/4=3.75<4$ then I know $\frac{4}{4n-3} > 4/4n=1/n$ since $4n-3
tinglingling
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1 answer

Help with pointwise convergence

for a function $g_n:[0,5] \rightarrow \mathbb{R}$ where $g_n=-nx^2+5\; if\; 0\le x \le 1/n$ $g_n=0\; if\; 1/n < x \le 5$ show pointwise convergence My attempt: I am very new to this subject but I thought for pointwise convergence I take…
JJd
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2 answers

How to solve this convergence problem?

$$ (\lim_{n \to \infty} {\sum_{k = 1}^{n} {\|{x_k}\|} < \infty} ) \Longrightarrow (\lim_{n \to \infty} {\sum_{k = 1}^{n} {\|{x_k}\|}^2 < \infty} ) $$
chris
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1 answer

convergence at a specific value

I have a series $S_n=\sum_{i=1}^{n} a*(1-i)$ where $a$ is an unknown constant independent of $i$. Is there a way to figure out for which $n$ the above expression converges to the value 0.01? After some calculations I derived $S_n=a*n-a*n*(n+1)/2$ If…
user148939
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Infinite sequence of real numbers converging to x and y

So the question is: Suppose $x_i$ and $y_i$ are infinite sequences of real numbers converging to x and y. Show that $(x_i + y_i)$ converges to $x+y$. Show that $x_iy_i$ converges to $xy$. Here's what I tried for $x_iy_i$: $|x_i - x| < \epsilon$…
Zoë Soriano
  • 181
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1 answer

Showing convergence of and evaluating an integral.

I want to show that the following integral is convergent, and evaluate it: $$\int_0^{\pi/2} \dfrac{1}{7 + \tan(x)} dx$$. I plugged the limit into mathematica and got "Directed Infinity". Tried the trick of multiplying the integral by $1$ and see if…
Tibz
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1 answer

Is there a closed form statement to describe every convergent series?

Curious whether the partial sum of every convergent series can be stated as a closed form expression. I saw this previous question, but they use the terminology "closed form value" and I'm not sure I understand what they mean. By convergent series I…
Connor McCormick
  • 101
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2 answers

Hints? Convergence behaviour.

I tried many things, including ratio test, root test, comparison test, cauchy criterion, rewriting w.r.t e^n and doing all these tests again, but no luck in any direction. Any hints? It seems most likely that I missed out on rewriting this stuff to…
nabu1227
  • 879
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1 answer

show that $f (x)$ is defined for all values of x

Sea $f(x)=∑_{n=1} ^∞$ $\frac{x^n}{n^n}$ Show that a) $f(x)$ is defined for all values ​​of x. b) evaluate the approximate form, where necessary $f(0),f(1),f'(0),f'(1),f''(0).$ c) obtain the MacLaurin series for $f'(x),f''(x)$. When applying the root…
taniadiaz
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3 answers

convergence or divergence of infinite rational series

Finding whether the series $$\sum^{\infty}_{k=0}\frac{5k^2+7}{8k^2+2}$$ is converges or diverges. What i Try: I am Trying to solve it using ratio test Let $\displaystyle a_{k}=\frac{5k^2+7}{8k^2+2}$. Then $\displaystyle…
jacky
  • 5,194
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1 answer

Finding convergence or divergence of series $\sum^{\infty}_{k=0}(-1)^{k}\frac{k}{3k-1}$

Finding convergence or divergence of series $$\sum^{\infty}_{k=0}(-1)^{k}\frac{k}{3k-1}$$ What i try:I am trying to solve it using Leibniz test Let $\displaystyle a_{k}=\frac{k}{3k-1}$. Then $$a_{k+1}
jacky
  • 5,194
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1 answer

test the convergency of $\arctan(k)$ series

Test the convergence of series $\sum^{\infty}_{k=1}(-1)^{k-1}\arctan(3k)$ What i have done is As we know that $$\arctan(x)<\frac{\pi}{2}\forall…
jacky
  • 5,194
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3 answers

Convergence of a nested sequence of sets that all contain a common set

I spent hours looking for answers to my question, but I could not find anything. I am looking for a proof that a nested sequence of sets $A_{n+1} \subseteq A_n \subseteq ... \subseteq A_0$ that all contain a nonempty set B ($B \subseteq A_n, \forall…
Quentin PLOUSSARD
  • 15
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2 answers

Finding convergence for an infinite sum

$$\sum_{k=1}^\infty(\sqrt[k] k - 1)^{2k}$$ I have to find convergence using the tests I know. (divergence,integral,ratio,root,comparison,limit comparison) My issue is I can't figure out how to not get an inconclusive test result.
Rut
  • 11
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2 answers

Recurrence that converge to $\pi/2$

I've saw this kind of the question and I can't find the way to prove. I want to know how the step of proving $x_n = x_{n-1} + \cos (x_{n-1} ) $ converges to $\frac{\pi}2$. The $x_1$ equals to 1.
TARDIS
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