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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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$a^{n^2}x^n$ which convergence test?

I've been trying to test $a^{n^2}x^n$ for convergence with the ratio, root, leibniz and direct comparison tests but don't seem to get loose of the $n$. any hints?
dan01
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Convergence of sequence b

Let $b_n$ be a sequence such that $b_{n+1}=\frac{b_n^2+1}{b_n} \ , \ b_1>0$ Is this sequence converging? explain. I managed to find that the series is monotically incresing, but couldn't show it is not bounded, thus not converging. Thanks!
Galc127
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Convergence of sum and multiplication

Iv'e got a question and I find it difficult to me. Let $ a_n,b_n$ be sequences. Assume that $a_n+b_n$ converges. Is $a_n\cdot{b_n}$ essentially converges? Prove. Thank you!
Galc127
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If $X_n\rightarrow X$ almost surely and $Y_n\rightarrow Y$ a.s., then is $X_n/Y_n\rightarrow X/Y$ almost surely true?

If $X_n\rightarrow X$ almost surely and $Y_n\rightarrow Y$ a.s., then is $X_n/Y_n\rightarrow X/Y$ almost surely true? Is there a theorem for this or is this not correct?
lightfish
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Proving that $\cos(n^a t)$ doesn't converge to $1$

Looking at the graph of the functions $\cos(n^a t)$ ($a>0$) it looks obvious that they don't converge to $1$ as $n \rightarrow \infty$. For integer odd $a$ it's easy to prove this directly, as $\cos(n^a \pi/2)=\cos(m \pi/2)=0$ for some odd $m$ in…
aplavin
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Weak convergence and convex hull

Let $\Omega$ be a bounded open set in $R^{n}$ and $K$ compact in $R^{m}$. Consider $L^{p}(\Omega,R^{m})$ defined as a vector valued $f=(f_{1},\cdots,f_{m})$ where $f_{i}\in L^{p}$ . Let $v_{j}: \Omega \to R^{m}$ be such that $$ v_{j} \to^{\star} v \…
user105554
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Show equivalence of convergence for different metrics

The problem words: Let $f: \mathbb{R}\rightarrow \mathbb{R}, x \mapsto \frac{x}{1+\left | x \right |}$ and let $d(x,y):=\left | f(x)-f(y)\right |$ be a metric on $\mathbb{R}$. Show that $x_n\overset{n}{\rightarrow}0\Leftrightarrow…
Joey
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Infinite Series Convergence using comparison test

I am trying to use the comparison test to determine whether the following infinite series converges. $$\sum_{n=1}^\infty \frac{1}{\sqrt{n^3+2n-1}}$$ $$\frac{1}{\sqrt{n^3}} > \frac{1}{\sqrt{n^3+2n-1}} $$ Is there a way to show that $1/\sqrt{n^3}$…
Quaxton Hale
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Convergence Proof!

The truth is, I don't even know where to begin with this question... Can someone please offer their assistance?
Vladimir Nabokov
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Proof of Convergence

$$\lim_{n\to\infty}\frac{5\cdot 2^n-4}{2^n-1}=5.$$ I have already proved this just using the definition of convergence. How do I go about proving this only using the sandwich theorem and sum/product/quotient rule? I can divide the whole expression…
Vladimir Nabokov
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Radius and Interval of Convergence Question

I have two problems in which I'm stuck finding the radius and interval of convergence: 1) $\sum\limits_{n=1}^\infty\frac{n^3(x+4)^n}{4^nn^{11/3}}$ Applying the ratio rule allows me to simply as…
user95297
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Convergence of $n^{n-1}/n!$

I need to check for convergence here. Why is every example completely different and nothing which I've learned from the examples before helps me in the next? Hate it! $$ \sum_{n\geq1} \frac{n^{n-1}}{n!} $$ so I tried the ratio test and I got $$…
loop
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How is the rate of convergence affected after a function transformation is applied?

Say we know that a sequence $x_t$ converges to $x$ at the rate $O(1/log t)$. Can I say at what rate $\exp(x_t)$ converges to $\exp(x)$ ? It comes as a subproblem in my work, and honestly I have no idea how to proceed or where to look. The…
chillar007
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Help with proof of convergence to Geo distribution

Updated for extra context and clarity I have this question that I can’t figure out and would appreciate some help. The first part of the exercise what about distinguishable particles and the co vergence to Poisson(r) was clear. However for this…
JenniferK
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Proof by contradiction that the sequence of terms tends to zero if the series converges?

I know how you can prove that the sequence of terms of a convergent series tend to zero by writing the term as the difference of series to $n$ and $n-1$. However, I would specifically like to know how to prove this via contrapositive/contradiction…
Max123456789
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