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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 on infite summation

given equation is convergent if k>1. $$ \sum_{n=1}^\infty \frac{1}{n^k} $$. can somebody tell me that how can i prove? My intuition: For k=1 $$\log(1+x) = x - \frac{x^2}2 + \frac{x^3}3 - \frac{x^4}4 ....$$ if we substitute x =…
weedfarmer
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Absolute convergence a criterion for unconditional convergence.

I'm working through Dr. Pete Clark's convergence notes here: http://alpha.math.uga.edu/~pete/convergence.pdf and I've been thinking about Exercise 3.2.2 (b), The question states that for the set $S = \mathbb{Z}^{+}$, the series $\sum_{i\in S}x_{i}$…
roo
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Prove that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$

Could someone please show me the proof that $\frac{2^{x+1}+(x+1)^2}{2^x+x^2}\rightarrow 2$ as $x \rightarrow \infty$ I have no idea where to begin with this one. Thanks.
BLAZE
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pointwise convergence of series?

We have just been introduced to infinite series of functions, and immediately started working on uniform convergence and Weierstrass' M-test, and how the sum of the series behaves in terms of integration, differentiation, etc. But what about…
Mikalea Danielsen
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Is this method for showing convergence legit?

I am working with some comparison testing for the first time, and am unsure if the method I have adopted is legit. For example, imagine I want to compare some improper integrals (or series) with the integral (or series) of $\frac{1}{x^P}$. We know…
Mikale Larson
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Using Comparison test the series for convergence. Stuck with understanding the hint.

I'm currently having some trouble trying to start this question. I'm meant to test the series $\sum_{k=1}^{\infty} \frac{k^3}{3^k}$ for convergence. The question provided a hint to start off by showing $\lim_{x\to\infty}x^pe^{-ax} = 0$ and use this…
user131516
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Convergence of a sequence and another sequence is bounded

Prove: If the sequence $$ converges to $0$, and the sequence $$ is bounded, then the sequence $$ also converges to $0$. Let $\epsilon > 0$, since $\lim_{n \to \infty} = 0$ applying the definition of the limit…
Wolfy
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Convergence of sequences

Prove: If the sequence $$ converges to $b\in \mathbb{R}$, then the sequence $<|a_{n} - b|>$ converges to $0$. Since $$ converges to $b\in \mathbb{R}$, denoted by $\lim_{n\rightarrow \infty} = b$ then for every $\epsilon > 0$…
Wolfy
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need help with this exercise about convergence

$g_n:I \rightarrow \mathbb R$ for $I\in [0,1]$, $g_n(a)=a^n(1-a)$ show $g_n$ converges pointwise. does it converge uniformly (proof) My attempt: I am pretty confuse. I am try to understand the difference between pointwise and uniform…
JBF
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Convergence of a recursive sequence of functions

Consider the sequence of functions $f_n:[0,1]\rightarrow[0,1]$ defined recursively: $$f_n(p)= 1-p + p (f_{n-1}(p))^2 \quad f_0(p)=1-p \quad f_n(1)=0$$ Computationally one can check that $\{f_n(p)\}$ converges to "a" solution of the characteristic…
typesanitizer
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What is the limit value of this function (sin..)

I wanted to ask you a question about this specific function: $$\lim\limits_{x\to\infty}\left( \sin(\sqrt{x})-\sin(\sqrt{x+1})\right)$$ Somehow I can't comprehend how to do this task On one hand sin has its specific attributes like jumping around $1$…
Kusottare
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Absolute convergence2

I have to test the series for absolute and conditional convergence $\sum_{n=2}^{\infty}$ $\frac{(-1)^{n-1}}{n^2+(-1)^n}$ $Notes :$ For absolute convergence def. I have $\vert \frac{(-1)^{n-1}}{n^2+(-1)^n} \vert$ If this convergence then the original…
tommy
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convergence test

I have to test if the series is absolute convergence and conditional convergence $\sum_{n=1}^{\infty}$$\frac{(-1)^{n-1}n}{(n+1)^2}$ This what I have so far: Im going to test for absolute convergence and if its fail then it would be conditional…
tommy
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Prove that $(y_n)$ converges

Prove that $(y_n) = \frac1n\sin({n\pi\over3})$ converges Now I know my RTP: ($\forall\epsilon\gt0)(\exists k \in N)(\forall n \gt k) \\ |(y_n)-c| \lt \epsilon $ but from there i get stuck.
user196606
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convergence series geometric test

Prove if this converges: $$\sum_{n=1}^\infty \frac{2^n+3}{3^n-1}$$ pf: using geometric $$0 < \frac{2^n+3}{3^n-1} \leq \frac{2^n + 2 \times 2^n}{3^n-\frac{3^n}{2}} = \cdots $$ and so on I know how to do the rest but my question is that where in the…
tom
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