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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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Does Cauchy condensation test relate to order of convergence/divergence?

For example harmonic series corresponds to Cauchy condensed series of $1+1+1+\cdots$ and since the Cauchy condensation is exponential, it just seems natural to reverse of it being related to order of convergence/divergence, which as it happens in…
jimjim
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How do I find explicit formula for $(x_n,y_n)$ and show that the sequence tends to converge to $(0,0)$?

How do I find explicit formula for $(x_n,y_n)$ and show that the sequence tends to converge to $(0,0)$? In $\mathbb{R}^2$, the sequence $(x_n,y_n)$, $n\in \mathbb{N}_0$ is recursively defined: $\begin{pmatrix}x_{n+1}\\ y_{n+1}\end{pmatrix}=\left…
Laja
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Find all x $\in$ R for which $\sum_{n=1}^\infty(1+\frac 12+...+\frac 1n)\frac {\sin nx} n$ converges.

Find all $x \in \Bbb{R}$ for which $$\sum_{n=1}^\infty\Big(1+\frac 12+...+\frac 1n\Big)\frac {\sin nx} n$$ converges. In the beginning, I want to apply Dirichlet Test on it. But I am still not sure that I've found all $x \in \Bbb{R}$.
Yao Yu Hsieh
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Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$ Now, by D'Alemberts Ratio test that implies (for convergence): $\lim_{n\to\infty} \lvert \frac{(n+1)!(2x-1)(2x-1)^n}{n!(2x-1)^n} \rvert<1$ $\lim_{n\to\infty} \lvert (n+1)(2x-1)\rvert<1…
user2250537
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Lower bound for $2+ \cos(t)$

I was doing this question on convergence of improper integrals where in our book they have used the fact that $2+ \cos(t) \ge1$. Can somebody prove this?
Geeta Negi
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Converge? $\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $

Determine whether the series converges: $$\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $$ I tired to use direct comparison test but since 1/k is not convergent, so I am not sure about in this case.
Nhay
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Convergent sequences out of bounded sequences

Let us consider a bounded sequence $\{a_n\}$ . Now as it is a bounded sequence it must contain a convergent sub-sequence, $\{b_n\}$. Now let us filter out $\{b_n\}$ out of $\{a_n\}$. As such we are still left with a bounded sequence. But , I want…
itp dusra
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Measure Theory - understanding usage of Dominated Convergence Theorem?

I am trying to understand the given proof below. But I don't really understand how the Theorem of Dominated convergence is applied? Which is the function that "dominates" the sequence? and we need to know that this funcion is Lebesgue integrable.…
Dan
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Alternating Series Convergence Test $(-1)^k [e - (1+\frac{1}{k})^k$

Determine if the following alternating series converge. $\sum_{k=1}^\infty (-1)^k [e - (1+\frac{1}{k})^k$] I know that $e - (1+\frac{1}{k})^k$ is positive for all x and can see that it is a decreasing function by comparing $e -…
GaramMasala
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Point-wise convergence of $\lim_{n\rightarrow \infty} \frac{x^n}{1+x^n}$

$\lim_{n\rightarrow \infty} \frac{x^n}{1+x^n}$ converges pointwise to $0$, if $0≤x<1$. $\frac{1}{2}$, if $x=1$. $1$, if $x>1$ Which is seen by checking the conditions first for $\lim_{n\rightarrow \infty} x_n$ and then for $\lim_{n\rightarrow…
mavavilj
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Convergence of infinite series.

I tried finding it's convergence by converting it into exponential series using first by ratio test and then by Raabe's test and reached closer to the answer but not the exact answer. $$\frac{a+x}{1!} + \frac{(a+2x)^2}{2!} + \frac{(a+3x)^3}{3!} +…
Surya Prakash Singh
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What's the largest term in a converging series?

Although it's quite a trivial question but is it always that the very first term in a converging series is the largest term of all the terms in that series ? Since if $\sum A_n$ converges, that is if $\sum A_n = a_0 + a_1 + a_2 +......$ converges…
Arnav Das
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Functions Preserve "ultra-ness" of Prefilters

I'm trying to prove the following proposition: Let $f:X\to Y$ be a map of topological spaces. Let $F$ be a prefilter on $X$. Then $(a)$ $f(F)$ is a prefilter on $Y$. $(b)$ If $F$ is an ultra prefilter, then so is $f(F)$. Note: Relevant…
roo
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Definition of an ultra pre-filter

I have come across the term "ultra prefilter" which has two possible definitions (that I can think of). I tried googling this first I swear! (but google thinks I'm looking for filtered water or a water purifier) The two most obvious meanings to…
roo
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If $\sum\limits_{n=0}^\infty A_n$ converges then $\sum\limits_{n=0}^\infty (A_n - A_{n+1})$ converges and its sum is $A_0$

I faced this question where it was already given that $\sum A_n$ is converging and I had to prove that $\sum (A_n - A_{n+1})$ is also convergent to the value $A_0$. I proceeded assuming $A_n > A_{n+1}$ from $\lim [A_{n+1}/A_n] <1$ as it was given…
Arnav Das
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