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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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What is the interval of convergence for $\sum \frac {\ln^3 n}{n^p}$

I need to find the right interval to make sum $\large{\frac{\ln^3n}{n^p}}$ converge. I guess that p>1, still I don’t know how to prove it. I tried using integrals and Taylor series, no success though.
Rose
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Convergence of indicator function of an interval

Suppose I have a continuous real random variable $X$ and an indicator function $f(X) = \textbf{1}_{\{a \le X \le b\}}$ where $c = (a,b) \in \mathcal{A} = \{(x,y) \in \mathbb{R}^2: x
eulerup
  • 75
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Convergence of $\left\{Y_n\right\}=\left\{(\frac{1}{n}+1,\frac{2}{n}+2)\right\}$

Is $\left\{Y_n\right\}=\left\{(\frac{1}{n}+1,\frac{2}{n}+2)\right\}$ is a convergent sequence? I'm studying through Takayama - Math Economics. I have to consider an open ball $B_r(x_o)\subseteq V$ and show that $\left\{Y_n\right\}\in V$, for $n>…
Muradin Bronzebeard
  • 339
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Convergence of $\prod_{n=1}^\infty n\sin\frac{x}{n}$

If $\prod_{n=1}^\infty n\sin\frac{x}{n}$ converges to a nonzero value, then $x=1$. The other direction is done using Taylor series, but how do I prove this?
eloquos
  • 1
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Convergence on variants of the Harmonic series

I am aware that the harmonic series diverges. I am also aware that the series $$\sum_{n=1}^\infty\frac{1}{p_n}$$ diverges. But why does the series $$\sum\frac{1}{p_{p_n}}$$ converge?
dahaka5
  • 275
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Number of iterations to achieve the desired convergence accuracy

For a small constant $\varepsilon>0$ and positive integer $n$, we are given a sequence of $(0,1)$-reals $\left\{a_0,a_1,\dots\right\}$, with $a_0=n^{-\varepsilon}$ and $$ a_{i+1}=\left(\frac{a_i}{1-a_i}\right)^2. $$ The question is about…
Matjaž Krnc
  • 446
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question on convergence

I'm working on a question I appreciate if you can guide me on how to solve it. Consider $X_j$ as integer random variables. We know $Pr(X_j = k) \rightarrow Pr(X = k)$ for every integer k. We also know that X is an integer-valued random…
user52144
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How to prove the convergence of the sequence?

I need to prove the convergence on $\mathbb R$: $$x_n = \frac{(2n)!!}{(2n+1)!!}$$ I don`t even know from what to start. Help me please.
PashaMinigun
  • 175
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Determine total expendature with constant return

Let's say I have a resource, call it E. There are four important values Em - the maximum amount of E we can have Ec - the current amount of E Er - the rate at which E replenishes (this is in % of Em) Ea - the cost amount in E of an action Let us…
corsiKa
  • 586
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Does convergence of $\frac{1}{N}\sum_{n=1}^{N}a_n$ imply $a_n/n$ converges to zero?

If for $a_n \geq 0$, the mean $\frac{1}{N}\sum_{n=1}^{N}a_n$ converges, then does $\lim_{n \rightarrow \infty} \frac{a_n}{n} = 0$?
Rajat
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Evaluate my proof on convergence and accumulation points

First of all, I define accumulation point as: Let $a$ be an accumulation point of $A$. $\forall \ \epsilon > 0$, $B_\epsilon(a) \setminus \{a\} $ contains an element of $A$. T/F question: If $a$ is an accumulation point of $A \subset \mathbb{R}$,…
user1691278
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convergence of $\sum_{n=1}^{\infty} \frac{1}{(n+1)^2 -1}$

How can i check convergence of $$\sum_{n=1}^{\infty} \frac{1}{(n+1)^2 -1}$$ ? How don't sure how to check it. I tried some of the tests.
user510010
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process convergence

Let (Ω,F,P) be a probability space and (Xn) ⊂ L1(P) and (Yn) be sequences of positive random variables which are adapted to a filtration (Fn). Suppose that the infinite sum: Y1+Y2+...< ∞ and E[Xn+1|Fn] ≤ (1 + Yn)*Xn, ∀n ∈N0, P-a.e. And I have to…
jessi22
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If $\hat{\theta_1}$ is consistent, and $\hat{\theta_1}-\hat{\theta_2}$ converge in probability, is $\hat{\theta_2}$ also consistent?

I'm trying to describe the consistency of $\hat{\theta_2}$. It is given that $\hat{\theta_1}$ ~ AN($\hat{\theta_0}$, A1/n). From this, we know that $\hat{\theta_1}$ is a consistent estimator of $\hat{\theta_0}$ (by Chebychev's Inequality). What can…
Jess G
  • 3
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Find the order of convergence and rate of convergence

Consider the sequence $ \ \large \ a_n=\frac{1}{4^{\left\lfloor \frac{n}{2} \right\rfloor}+(k+1) \ mod \ 2} \ $ . Find the order of convergence and rate of convergence . Answer: From the definition of the sequence the first few terms are $ 1/2,1,…
MAS
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