# Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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### Cannot prove these two statements about Bernoulli sequence

In the 8th chapter of An Introduction to Probability Theory and its Applications. Vol 1 (by William Feller) there are two problems that I'm stuck with, and since the two are interconnected, I decided ...
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### How do we know this series diverges?

I’m trying to solve some problems relating to convergence, and I came across this problem: $$\sum_{i=2}^{\infty} \frac{1}{(\log (\log i))^{1+\varepsilon}} = +\infty , \,\,\, \varepsilon > 0$$ My ...
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### Proof a sequence is convergent

Given 2 sequences $\{x_n\}$, $\{y_n\}$ such that: $${y_n^{2} \leq \frac{1}{n} + {x_n}{y_n} \sqrt{x_n}}$$ with ${\forall} n \in$ $\mathbb{N}$. Suppose that ${x_n} \to 0$. Prove that $\{y_n\}$ ...
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### Does the given sequence converges or diverges? [closed]

If b>1 and is a real number then does the sequence converge or diverge. I intuit that it diverges because b^n grows faster than (n!)^1/2?
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### Just as Fibonacci addition approaches phi with a(n-1)*phi = a(n), Fibonacci multiplication approaches phi with a(n-1)^phi = a(n). Can we extend?

Fibonacci addition converges on the golden ratio between consecutive values, i.e. a(n)/a(n-1) = phi. Example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... 55/34 rounds to 1.618. I realized today that Fibonacci ...
1 vote
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### Order of Convergence in Simulations [closed]

I have made a simulation of Buffon's Needle Problem and I am trying to show that it converges, however, when I try estimating the order of convergence using Euler's method I get different answers each ...
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### Why does this sum not diverge?

So I'm asking this question in general but with a motivating example: Say we have a function $$f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$ So both the top ($x^n$) and bottom ($n!$) of this function grow -...
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### Convergence and sum of the series $\sum\left(L-a_n\right)^2$, where $a_0=0$ and $a_{n+1}=(a_n)^2+\frac{1}{4}$ and $L=\lim\limits_{n\to\infty} a_n$

I am trying to find out whether the series $\sum\left(L-a_n\right)^2$ converges (where $\{a_n\}$ is the sequence defined by the recurrence relation $a_0=0$ and $a_{n+1}=(a_n)^2+\frac{1}{4}$ and $L$ is ...
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### Convergence of the Infinite Series of Reciprocals of Primes to 1

I am a junior high school student in Japan and I came up with an idea for something, but I don't know the name of it at all, even though I looked it up. Furthermore, I know only a little about high ...
1 vote
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### Show convergence of a sequence using $\varepsilon$-definition

I would like to show that the sequence $\frac{4n^2+3n}{n^2-4n+4}$ converges to $4$ as $n \to \infty$ by using the $\varepsilon$-definition of convergence. I am aware that it would probably be easier ...
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### Find convergence a.s and in probability of $X_n$ if $P(X_n = 0) = \frac{1}{2n^\lambda}$ and $P(X_n = 2) = 1 - P(X_n=0)$

Given $\lambda > 0$ I want to find the convergence a.s and the convergence in probability of $X_n$ if $P(X_n = 0) = \frac{1}{2n^\lambda}$ and $P(X_n = 2) = 1 - P(X_n=0)$. For the convergence a.s my ...
### Given $(U_n)_n$ a sequence of i.i.d random variables $U[0,1]$, find convergence of $1_{(0,1/n)}(U_1)$ and $1_{(0,1/n)}(U_n)$
Given $(U_n)_n$ a sequence of i.i.d random variables $U[0,1]$. I want to find the convergence in probability and a.s of the sequences $X_n = 1_{(0,1/n)}(U_1)$ and $Y_n = 1_{(0,1/n)}(U_n)$. I don't ...