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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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Is power series expansion not allowed for Gauss' Convergence Test?

Given the following Legendre series, I wanted to test its convergence at $x^2=1$ when $l$ is not an integer \begin{align} S = \sum_{j=0}^{\infty} u_j && u_{j+2} = \dfrac{(j+1)(j+2)-l(l+1)}{(j+2)(j+3)} x^2 u_{j} \end{align} I tried using Gauss' test…
linuxfreebird
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real sequence and convergence in probability

$X_n$ is a sequence of random variables.$X_n \equiv a_n$, $a_n $ is a real sequence. Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n \to \lim_{n\to\infty} a_n$ in probability. I have a feeling that the above…
kris91
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Does from this follow that the sequence $(f_n)$ converges uniformly to $f$?

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions that are uniformly continious on a compact set $K$ and that converges pointwise to a function $f$ that is uniformly continious on $K$, too. Can I know from this that the sequence converges…
mathfemi
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Convergence of $\sum_{n=0}^{\infty}ne^{-\beta n}$

I am working in a problem of statistical mechanics and a partition function I found is of the form: $$ Z = \sum_{n=0}^{\infty}(n+1)e^{-\beta n} $$ I used the ratio test and the series converges. I want to find a closed form for this. I…
Thiago
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End points interval of convergence

I am trying to evaluate the end points of an interval of convergence of the series: $$\sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}}$$ Applying root test: $$\lim_{n \to \infty}\sqrt[n]{\left|\frac{(n+1)x)^n}{n^{n+1}}\right|}$$ $$=\lim_{n \to…
Johnmgee
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What is the probability limit and limit distribution of the estimators given that$ X_i$ are iid

This is more of a practice question but I'm not sure how to really proceed. Say that $E(X)=0$ and $Var(X)=\sigma^2$. Firstly I am required to find the probability limit of the estimator as $n$ go to infinity. I tried to break it up into I know…
user131516
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Power Series -- Radius of Convergence and Interval of Convergence

Find both the radius of convergence and interval of convergence of the power series. $$\sum_n\frac{(-1)^n\ln n}{\sqrt{n}}(x-2)^n$$ I believe that the radius of convergence is $R=1$. But I'm having a hard time with this entire question. the $\ln n$…
jerry2144
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Convergence of an iteration

Considering the following iteration: For any initial value of $x^{(0)}$, find the value of $\alpha$ for which the iteration converges.
user146402
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Radius of convergence of series

Find the radius of convergence of this series: $$f(x)= \sum_{j=1}^{\infty} \ \frac{(-1)^{j-1}}{j}(x-1)^j$$ I'm not sure what test to use to get the necessary result. I tried using the root test, but got an expression with both x and j that I can't…
kiwifruit
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Showing uniform convergence of series

Show that $\displaystyle \sum_{j=1}^{\infty} \frac{-2j}{(x^2 + j^2)^2}$ converges uniformly. Don't know how to do this problem since $x$ and $j$ are in the expression together. Is there a convergence test I can use?
kiwifruit
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Series convergence or divergence

Determine whether this series converges: $\Sigma e^{-j+sinj}$ I know that this series is $\leq$ than $\Sigma e^{-j+1}$, but I am having trouble getting this in a form appropriate for a convergence test...
kiwifruit
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Prove that $\varphi^n(t) \rightarrow 0$ when $n \rightarrow \infty$

I have to prove a lemma: If $\varphi: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ is monotone increasing and $\varphi(t) < t, \ \forall t \in \mathbb{R}_+$, then $\varphi^n(t) \rightarrow 0$, $(n \rightarrow \infty)$. To be clear, $\varphi^2(t) =…
Egor N
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what is the Convergence radius and what happen at the edges?

What is the Convergence radius and what happen at the edges? $$\sum_{n=1}^{\infty}\frac{(x+2)^{n^2}}{n^n}$$ Thank you
A student
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Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that?

Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that? I was trying to prove Mertens third theorem and i got stuck at this.
esege
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Pointwise, and uniform convergence

Suppose $f$ is continuous on $[0,1]$ and $f_n(x) = f(x)^n$. Explain why this sequence does not converge uniformly when $f(x)=1$? When $f(x)=1$, $f_n(x)$ is $1$ for all $n$. But I don't see why this means there is no uniform convergence. Can't we say…
user138602
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