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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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$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges

Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges. I have looked at an answer here, but I do not understand what these results give us. For example, in the first answer: $$\frac{2^j + j}{3^j - j} \le \frac{2^j + 2^j}{3^j…
kiwifruit
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Does differentiability imply convergence

Can we say that if the limit of a sequence of functions is differentiable then the sequence is convergent? I mean, I know that $\frac{\partial f(x,t)}{\partial x}$ exists. If I specify a sequence based on the definition of the derivative, something…
The Mighty Algernon
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What is the convergence or divergence of $\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$

$\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$ Can you show me the work for this question
Lindsay Sinai
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How to show $\Gamma (n)$ is convergent if and only $n>0$?

How to show $\Gamma (n)$ is convergent if and only $n>0$ where $\Gamma (n)$ is the gamma function.
esege
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Find the radius of convergence and the interval of convergence of $\sum_{n=1}^{\infty}\frac{(2n)!}{n!}x^{3n}$.

Find the radius of convergence and interval of convergence for the following summation: $$\sum_{n=1}^{\infty}\frac{(2n)!}{n!}x^{3n}$$ I don't know how to deal with the $(2n)!$ in this question. Any help is appreciated!
user138069
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Show that $\lim _{n \to \infty}\left (a_{n} + b_{n}\right) = a + b$ help ??

Assume that $\lim_{n \to \infty} a_n=a$ and $\lim_{n \to \infty} b_n=b$ Show that $\lim_{n \to \infty}(a_n+b_n)=a+b$ and that $\lim_{n \to \infty} a_nb_n=ab$ by using the definition of convergence of sequences (well this is two seperate questions,…
teggreg
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Convergence and Divergence

Suppose that the series $\sum_{n=1}^\infty a_n$ is conditionally convergent. Prove that the series $\sum_{n=1}^\infty n^2a_n$ is divergent. How should I start to prove this? I have absolutely no idea how to go about solving this problem. Thanks in…
Haxify
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Bounded function

${f_n}$ is a sequence of continuous functions on $\Bbb R$, and $f_n \rightarrow f$ uniformly on every finite interval $[a,b]$. If each $f_n$ is bounded, is it true that $f$ must be bounded?
kiwifruit
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Convergence on interval vs on R

If $f_n$ -> f uniformly on every finite interval [a,b], does this imply it converges uniformly on R? I would think yes, since R can be expressed as a sum of intervals...
kiwifruit
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Convergent sequence has one limit point

I need to prove that if a sequence is convergent, it has exactly one limit point. I have started proving this by contradiction, but can't seem to come to an equation that is obviously wrong. Please help.
user989
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Square of Cauchy sequence

Please help find an example to prove that the convergence of ${a_n^2}$ does not imply the convergence of $a_n$, where both are sequences.
user120494
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Sum convergence test (arctan involved)

I need to help with this problem: Find out if this sum is convergent or not. $\sum_{n=1}^\infty (arctan(n+1)-arctan(n))$ Thank's for help! :)
user122413
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Convergence of bounded sequence

Problem: $b_n$ is a bounded sequence and $a_n$ converges to 0. Prove that $a_n b_n -> 0$. I would understand how to do this proof if I knew that the sequence $b_n$ was convergent. But that is not the case in general? Is there a different method to…
user120494
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Bhaskara-Brouncker Algorithm Convergence rate

http://www.mathpath.org/Algor/squareroot/algor.bhaskara.brouncker.htm What is the convergence rate of this algorithm? I have tried various google searches to no avail. (For example, the convergence of Newton's Method is Quadratic, the convergence of…
user1380792
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Convergence/Divergence of sums

I was asked to determine if the next sums converge absolutely, converge conditionaly or diverge. For the first question I tried to use Leibniz: Define $a_n=\frac{1}{n^a ln(n)}$. It's easy to show that $a_n$ is decreasing. I found that if $a \geq…
Galc127
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