Mathematics Stack Exchange
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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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Proving divergence or convergence of the infinite sum of $\sum \frac{1}{{(\ln k)} ^{\ln \ln k}}$

I was trying to determine if the following infinite sum (from 1 to infinity) diverges or convergence, but I am completely stuck. What test would be most appropriate? I know that $\ln k
William
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Can someone please explain the significance of the subscript 2 on this picture?

[] Does the subscript here refer to the L2 norm? In the right hand side, are they saying that this is are squaring the square root of the difference in squares? Is convergence here reffering to the real minimum of the function "f"? Is this…
stats_noob
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For which power do these series converge for odd and even termes

Hi I really don’t know how to solve this question: For which a do these series converge ? $(\sum_{i=0}^n \frac{1}{i!})^a $ only if i is odd $\sum_{i=0}^n \frac{(2^i)}{i!}$ only if i is even I know the even serie converge to something smaller than…
Jouja
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How can the limit of convergence of a binomial series of the function $(1+x)^p$ be $|x|<1$?

For a function $(1+x)^p$, when we write the power series and use the ratio test to find out the interval of convergence, we get $|x|<1$. But take a positive value of $p$, 3 for example. The function becomes $(1+x)^3$ and surely the power series…
R Premjith
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How to prove $\sum_{n=1}^{\infty}a_nn^{\frac{1}{n}}$ is convergent if $\sum_{n=1}^{\infty}a_n$

How to prove that $\sum_{n=1}^{\infty} n^{\frac{1}{n}} a_n$ is convergent. Where $\sum_{n=1}^{\infty}a_n$ is convergent. I know that Abel's Test: Let $\sum_{n=1}^{\infty}a_n$ is convergent and the sequence $$ is monotonic and bounded…
Largest Prime
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How do I show that convergence by a measure implies L^1 convergence?

I have the following problem: Let $f,f_1,...,f_n,...\in L^1(\Omega)$. We assume that there exists $g\in L^1(\Omega)$ such that $|f_n|\leq g$ for all n. Show that $$f_n\stackrel{\mu}{\rightarrow}f\Rightarrow f_n\stackrel{L^1}{\rightarrow}f$$ How…
user123234
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Proof the convergence of the series $a_{n+1}=1-\frac{1}{2+a_{n}}$

I try to improve my writing style and I went from "doesn't make any sense" and "wrong", to "missing lots of steps here", until "verbose". Can anyone make suggestions whether the proof is correct in the first step and how to improve it? Prove that…
Iwan5050
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Convergence of 1d3-1

I have an infinitely large pantry. I put some number (say, 100) of special potatoes in there. Every day, each potato has an equally-likely probability of either: Dying Living Living plus producing a single identical clone of itself After…
Sarov
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Under what conditions it is possible to show $||x_{k+1}-x_*|| \leq ||x_k-x_*||$ implies $x_k \to x_*$?

Let $\{x_k\}_{k\geq 0}$ be a sequence such that $||x_{k+1}-x_*|| \leq ||x_k-x_*||$ holds for all $k=0, 1, \dots$ where $x_*, x_k \in \mathbb{R}^n$. Under what conditions it is guaranteed that $x_k\to x_*$? My try: I know when there is no equality we…
Saeed
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Find the taylor series for $\log(z+21)$ and then determine the radius of convergence using the ratio test

I am really stuck with this. I got a taylor series of ${((-1)^{(n+1)}z^n)}/{(21^n(n!))}$ at $z=0$ if $n \neq 0$; however, wolfram alpha gave me a different answer as attached. My intuition tells me that the taylor series converges in the circle…
Elli
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D'Alembert's ratio test inconclusive except when for a sufficiently large $n: u_{n+1} / u_n\geqslant1$.

My textbook and Wikipedia similarly state: Say $r=\liminf\limits_{n\to\infty}\,\left\lvert\dfrac{u_{n+1}}{u_n}\right\rvert\quad$ and $\quad R=\limsup\limits_{n\to\infty}\,\left\lvert\dfrac{u_{n+1}}{u_n}\right\rvert\;$. If $R < 1$, the series…
Erithax
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When do series converge and when do they diverge

So I have read that series converge when a limit exists, ant it does diverge if there is no limit or it goes to infinity. So then I saw this example, where I have series: And then 1) When a > 1 they say it converges, well which I understand why,…
Norbiuxx
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Is there any example such that $\lim\limits_{k \to \infty} S_{2k} < \infty$ but $\lim\limits_{k \to \infty} S_k = \infty$?

Let $S_k = \sum\limits_{n=1}^{k}a_n$ be a series. Is there any example that $\lim\limits_{k\rightarrow\infty}S_{2k}$ exists but $\sum\limits_{n=1}^{\infty}a_n$ is convergent.
Aseon
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Smooth convergent sequence

Suppose that you have a convergent sequence of real numbers $x_n$ and assume that you want to impose conditions so that this sequence converges to some limit $x^*$ for $n\rightarrow \infty$ in a smooth way that is without jumps. In other terms I…
user279687
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$\sum_{n=1}^{\infty}(A_n+B_n)=\sum_{n=1}^{\infty}A_n+\sum_{n=1}^{\infty}B_n $

If $\sum_{n=1}^{\infty}A_n$ is convergent and $\sum_{n=1}^{\infty}B_n$ is divergent then can we write $\sum_{n=1}^{\infty}(A_n+B_n)=\sum_{n=1}^{\infty}A_n+\sum_{n=1}^{\infty}B_n $?
user860738
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