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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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finding N in epsilon proof of convergence

I get what the epsilon-definition of convergences means but I still don't get how we find a fitting $N$. Let me explain what I mean with a little (simple) example: Let $a_n = \frac{n}{n+1}$, determine the limit and prove it. So, the limit is easy to…
JonDoeMaths
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Proof for bracketing a positive series

I've been asked to prove that given a series consisting of non-negative terms. By bracketing these terms in a particular way, I get a new series that converges. Prove that the original series converges also and to the same sum. I know that it is…
dahaka5
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Convergence of series n/2^(n-1)

I have to find the value of 3.9^1/2.27^1/4.81^1/8....upto infinity The series can be written as 3^£n/2^(n-1) where n is upto infinity. I solved the question and found that n/2^(n-1)=0 Is this correct? or I am doing wrong?
Sona23
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What is the sum of 1(1 - 1/3 ) - 1/2(1 - 1/3 + 1/5 - 1/7)+1/3(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11) - 1/4(...)...?

The series converges (conditionally) since $$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}+\ldots $$ converges to $\ln{2}$, and $$ 1 - \frac{1}{3},\quad 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7},\quad 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} +…
Graham Logan
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calculate the sum of $\sum_{n=1}^{\infty}b_n$

If we let $(a_n)$ and $(b_n)$ be sequences, and suppose there exists $N\in \Bbb N$ such that $a_n=b_n$ for all $n>N$. Suppose $\sum a_n$ converges and $\sum_{n=1}^{\infty}a_n=S$. I am looking to calculate the sum of $\sum_{n=1}^{\infty}b_n$ My…
user123
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How should I interpret $N(\epsilon)$ in the equation below?

The simplest definition of convergence implies that $X_n$ will converge to $X$ if for any arbitrary number, say $\epsilon > 0$, $|X_n-X|< \epsilon$ for all $n>N(\epsilon)$ so that $\lim_{n \to \infty} X_n=X$. How can I interpret $N(\epsilon)$? ...…
user91991993
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two convergent subsequences

Given is the following sequence $a_n:= (-1)^n*(\frac{241216}{n}+1)$ and I have to find two convergent subsequences with different limits. At first I thought of $a_{n_1}:= (-1)^{2n}*(\frac{241216}{2n}+1)$ and $a_{n_2}:=…
user337258
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Convergence that is slower than algebraic convergence

all, suppose that $$ \lim_{t\rightarrow \infty} r(t)=r_0>0, $$ I am wondering whether there exists $C>0$ and $\alpha>0$ such that $$ |r(t)-r_0|\leq C(t+1)^{-\alpha}, \quad \forall t>0. $$ If it were the case, then algebraic convergence is the…
Teh
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Definition of Divergent Sequence

I tried to check the definition of divergent sequences in many places. All of them define divergent sequence as a sequence which is not convergent. I wonder if a Cauchy sequence (which is not convergent in a non-complete normed space) is still…
Waroth Kuhirun
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Proof: convergence about power series

$\sum a_n x^n$ power series If the coefficients $a_n$ equal the answers of a family of counting problems indexed by n we call the power series a generating function for the counting problem. $1 + x + 2x^2 + 6x^3 + 24x^4 +$ equals the…
S.káham
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Convergence of a sum of polynomial

I have just started reading the book "Measure, Integral and Probability" 2nd ed. by Marek Capinski and Ekkehard Kopp. The book starts out with a discussion on the Riemann Integral, its scope and limitations. An example is given. The example results…
HenrikJson
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Should $a_n $ be bounded sequence

If $a_n $ is a sequence such that $\sum |a_n||q_n| < \, \infty$ whenever $\sum |q_n| < \, \infty$ then $a_n$ is a bounded sequence. Is this statement true? I used $a_n$ as $n^4$, and $q_n$ as $ n^{-8}$ for counterexample, is it correct? The…
jnyan
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Is monotonic and convergent series bounded by the limit?

Problem: Suppose that $X_1\leq X_2\leq\cdots $ and $X_n \stackrel{P}{\to} X$, is the series bounded such that $X_n \leq X$ for all $n$? Similar arguments are made in the first answer by Saz at Monotone increasing sequence of random variable that…
Sheldon
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Sequence of loops that converges to the unite circle with length greater than 2\pi

I would like to know how to construct a sequence of loops that converges to a circle in the Hausdorff distance, but has constant (or increasing) length greater than the circumference of the circle.
compaq
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Convergence of the sequence $\left(\frac{1}{n}\right)^{\sin\left(\frac{1}{n}\right)}$

I tried using the logarithmic method. But I couldn't work with it as I am not sure we can apply L'Hospital to the indeterminate form -infinity/infinity. Please help me prove the convergence.
Ayush Kumar
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