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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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convergence or divergence of $\sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}$

Finding whether the series $$\;\; \sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}$$ is converges or Diverges What i try We know that $\sin^2(x)\leq 1$ for all real number. So $$\sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}\geq…
jacky
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Convergence of $\sum^{\infty}_{k=1}k^2\tan\frac{k+2}{k^2+5}$

Convergence of series $$\sum^{\infty}_{k=1}k^2\tan\frac{k+2}{k^2+5}$$ What I have tried: using $\tan x>x$ for…
jacky
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convergence or divergence of series $\displaystyle \sum^{\infty}_{k=1}3^{-\ln(k)}$ using integral test

Finding convergence or divergence of series $\displaystyle \sum^{\infty}_{k=1}3^{-\ln(k)}$ using integral test What i try: let $f(x)=3^{-\ln(x)}$ Then $\ln(f(x))=-\ln(x)\cdot \ln(3).$ Then $\displaystyle…
jacky
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convergence of improper integral using comparison test

Convergence of Improper Integral $$\int^{\infty}_{0}\frac{2x}{(x^2+1)^3}dx$$ using comparison test What i try Put $x^2+1=t$ and $2xdx=dt$ and changing limits So integration is…
jacky
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Pro.5 sec.3.2 in kreyszngs "introductory Functional analysis with application

Show that for a sequence $(x_n)$ in an inner product space the conditions $\lVert x_n \rVert \to \lVert x \rVert$ and $\langle x_n,x \rangle \to \langle x, x\rangle$ imply convergence $x_n \to x$.
top 4u
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Convergent of Divergent of sequence having Rational terms

Finding whether the sequence $$a_{n}=\sqrt{1+n}\sqrt{2-n}-\bigg(\sqrt{1+n}\bigg)^2+2n+1$$ is convergent or Divergent. If convergent, Then $\lim_{n\rightarrow \infty}a_{n}$ equals What i try: $$\displaystyle \lim_{n\rightarrow \infty}…
jacky
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Convergence of a sequence of harmonic means

I'm stuck in this, I need some hint: Prove that if a given sequence $z_n$ with $n\ge1$ converges to a limit $L$ then $w_n=\frac{n}{\sum_{i=1}^n\frac{1}{z_i}}$ converges to the same limit. I tried a lot of things but it leads me nowhere. I just…
14max
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if a series cannot be evaluated through the root test algebraically, will it still have the same limit value as ratio test?

Theorem: convergence for the ratio test implies convergence for the root test. So whenever the ratio test works (i.e. tells you whether the series converges), the root test also works and the limits coincide. If…
user29418
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Convergent and divergent series

I am studying the behaviour of product of a convergent and a divergent infinite series. I found a example in which product series come out to be a convergent series . But can't get a divergent series Does product series always comes out to be…
Shubham singla
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Finding radius of convergence of series $\sum^{\infty}_{r=1}x^{r}\cdot \cos^2(r)$ is

Finding Radius of convergence of series $$\sum^{\infty}_{r=1}x^{r}\cdot \cos^2(r)$$ is What i try Let $a_{n}=\cos^2(n)\cdot x^{n}.$ Then $a_{n+1}=\cos^2(n+1)\cdot x^{n+1}$ Now $$\lim_{n\rightarrow \infty}\bigg|\frac{\cos^2(n+1)\cdot…
jacky
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Finding Convergence of series $\sum^{\infty}_{k=1}\frac{3k}{k^2+4}$

Finding Convergence of series $$\sum^{\infty}_{k=1}\frac{3k}{k^2+4}$$ using Integral test or Divergences Test. What i try Let $$\sum^{\infty}_{k=1}\frac{3k}{k^2+4}=3/5+6/8+\sum^{\infty}_{k=3}\frac{3k}{k^2+4}$$ Let $\displaystyle…
jacky
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evaluation of convergence of series with irrational terms

Evaluation of convergence of series $$\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}$$ using camparasion or limit Camparasion Test What i…
jacky
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convergence or Divergence of rational Irrational

How can i find whether the integral $$\int^{\infty}_{1}\frac{\sqrt{x}-1}{x^2+x+2}dx$$ in converge or diverges What i try I try to solve the Given Integal by substuting $x=t^2$ and $dx=2dt$ and changing limits And it convert into…
jacky
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convergence or divergence of rational series using camparasion or limit camparasion test

Using camparasion or limit camparasion test Finding whether the series $$\sum^{\infty}_{k=1}\frac{5k^9-k^5+8\sqrt{k}}{3k^{11}-k^4+2}$$ is converge or diverge. What i try…
jacky
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finding whether the divergence test is applicable on given sequence

Find whether the divergence test is applicable for sequence $$a_{n}=e^{-\frac{6}{n}}$$If applicable, then $\displaystyle \lim_{n\rightarrow \infty}a_{n}=$ What I've tried: I did not understand how the divergence test is applied here. But I am…
jacky
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