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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 .

21990 questions
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Limit in probability vs Almost sure limit

is it possible to create a sequence of random variables that converge to some constant in probability, but converge to infinity almost surely? This question is motivated by the fact that for every subdivision of the interval $[0,t]$ whose mesh tends…
johannes
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Series convergence

I have been given a rearrangement of the alternating harmonic series that follows the following…
dahaka5
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Pointwise limits in Real Analysis 4

In the following two questions I need to find the pointwise limit of the sequence defined for $x\in R$. I have had some success with the first example and believe I have done it correct. But in the second example I am not sure how to find the…
Bl409
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Rearrangements of the harmonic series

Trying to understand how rearrangements work. A very common example of rearrangements seems to be the alternating harmonic series, $$\sum _{n=1}^{\infty} \frac {(-1)^{n+1}}{n}$$ Plugging in values of $n$ gives, $$1-\frac{1}{2}+\frac {1}{3}-....$$…
mp12345
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Limit Comparison test: Selection of auxiliary series

How can we solve these questions using limit comparison test, specifically how can we choose $b_n$ for any series, (The $n^{th}$ terms are given) (1) $\dfrac{1.2.3.....n}{1.3.5....(2n-1)}$ (2) $\dfrac{(2n-1)}{n!}$ (3)…
Chin
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Calculating error in a series

I need to determine if the following series cponverges. Which I believe I have done correctly, but then I must determine how many terms must be summed to guarantee an error no greater than $\frac {1}{10}$. This second part I am having some trouble…
hburt
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Convergence of the sequence

Let $u_n=\frac{1}{(\log n)^{\log n}}$ then what can we say about convergence of the sequence and series as well... I tried Cauchy condensation formula but that doesn't seems to work...
Nitish
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Show that $y=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}$ converges for all $x \in \mathbb{R}$

I'm thinking of approaching this with the ratio test, $\frac{a_{n+1}}{a_n}=\frac{x}{n+1}$. I know that the limit of this ratio as n approaches infinity is zero, therefore the series must converge?
Chad
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Does $\sum_{k \ \text{odd}, \ k>0}^{N} \frac{\sin(kx)}{k}$ converge to $f(x)$ in $L^2$?

So I looked up that for something to converge in $L^2$ we must have that $$ \int_{I} |f_n(x) - f(x)|^2 dx \to 0 \text{ as } n \to \infty $$ With $$ f(x) = \begin{cases} -1, & x\in [-\pi, 0) \\ 1, & x\in [0, \pi]. \end{cases} $$ And $f_n(x) =…
Olba12
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Question about the convergence of the root test

According to the root test, we have that: $\lim_{n\to\infty} \sqrt[n]{|a_n|} \left\{\begin{align}&>1 \implies \text{divergent}\\&=1\implies \text{???}\\&<1 \implies \text{convergence}\end{align}\right.$ But why doesn't it hold when you take the…
Frank Vel
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Why is c.d.f. used in defining convergence in distribution.

The textbook of probability said: A sequence of random variables $X_1, X_2, X_3, \ldots$ converges in distribution to a random variable $X$, shown by $X_n \overset{d}{\to} X$, if $$ \lim_{n\to\infty}F_{X_n}(x) = F_X(x), $$ for all $x$ at which…
Danny_Kim
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Does $\sum_{k=1}^{\infty} \frac{1}{k\sqrt{\vphantom{} k+1}}$ converge?

Does the following series converge? $$\sum_{k=1}^{\infty} \frac{1}{k\sqrt{\vphantom{|} k+1}}$$ I tried using the ratio test and the comparison test but I wasn't able to solve this. I think I should try manipulating the denominator to use…
112358
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Convergence demonstration?

I got the following assigned as homework: "Demonstrate that the following series are convergent:" $$\sum_{k=0}^N a^k\\ \sum_{k=0}^\infty a^k \\ \sum_{k=0}^\infty ka^k \\\sum_{k=0}^\infty k(k-1)a^k\\ \sum_{k=0}^\infty k^2a^k$$ I know most of these…
Ana Ameer
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For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment?

For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment? To examine convergence I usually use rules for convergence, but what to do when I have intervall?
Alen
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$Y_n\xrightarrow{P}E(X_i)$

I have this problem at hand $X_1,X_2,\cdots $ are iid random variables with finite second moments.Define$$Y_n={2\over n(n+1)}\sum_{i=1}^n iX_i$$ Show that $Y_n\xrightarrow{P}E(X_1)$. I know that $\sum\limits_{i=1}^n {(X_i-\mu)\over…
Qwerty
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