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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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Exploring what happens at both ends of the interval of convergence

I have to find the radius of convergence and see what happens at both ends of the interval of convergence for the following power series: $$ \sum_{n=0}^{\infty}\frac{(3n)!}{n!(2n+1)!}(x-3)^n $$ I found the radius of convergence to be $\mid x - 3\mid…
Daniel
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Finding the radius of convergence from a closed form representation of a series

Say we have the following: $$\sum_{n=0}^\infty a_n = \frac{\sqrt{5-x}+45\sqrt{x}}{(5-x)^{\frac{a}{b}}} $$ I have made this up on the spot and by no means is this true. How would I go by finding the radius of convergence by the closed form? Do I just…
H5159
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Showing that $\sum\limits_{n=1}^{\infty}\frac{i^n}{\sqrt{n}}$ is convergent

I want to show that $$\sum\limits_{n=1}^{\infty}\frac{i^n}{\sqrt{n}}$$ is convergent, but not absolutely convergent. Demonstrating that it is not absolutely convergent is easy since $$\left|\frac{i^n}{\sqrt{n}} \right|=\frac{1}{\sqrt{n}}$$ but…
emka
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Completeness of set

Show that the set of bounded continuous functions is complete in the sup-norm. My work: we want every Cauchy sequence to have a limit in the same set. But I don't understand what we can start with. If we take some Cauchy sequence $||f_n - f_m|| \leq…
user136153
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Convergence proof of non convex formulation

Assume that we have a non-convex optimization $\min_{A,B} f(A,B)+\lambda g(A,B)$. Specifically, $f(A,B)+\lambda g(A,B)$ is not joint convex, but it is convex with regard to one variable when fixing another variable. Now the question is could we use…
Lia
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terms $(x^n/n!)$ approach 0 faster than $(1/x)^n$

Can anyone help me prove that the terms $\frac{x^n}{n!}$ approach $0$ faster than $(\frac{1}{x})^n$. Thank you in advance for the help.
user133993
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Convergence and bound

${f_n}$ is a sequence of continuous functions that converges uniformly on [0,1]. Show that there is an M such that $|f_n(x)|\leq M$ for all n and x. My thoughts: since the functions are continuous on a closed bounded interval, they must be bounded.…
kiwifruit
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Convergence = "closer and closer"

I am asked to find a sequence and a number such that $$|a_{n+1}-a| \lt |a_n-a|$$, but $a_n$ does not converge to a. Please help!
user120494
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Fastest converging method for calculating e

Does anyone know what the fastest converging equation is for calculating e? I have found many different equations when I have been searching google, but I was wondering which ones can yield the most digits of e in the fewest terms. Thank you.
Progo
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Fractions in convergence proof

Sequence: $a_n = \sqrt{2+ \frac{3}{n}}$ To prove convergence, want to show that $\left|\sqrt{2+ \frac{3}{n}} - \sqrt{2}\right| \le \varepsilon$ Simplifying, we get that $\sqrt{2+ \frac{3}{n}} - \sqrt{2}= \frac{3/n}{\sqrt{2 + \frac{3}{n}} +…
user120494
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How to show theoretical convergence

I was doing various tasks about convergence/divergence of series, where i had to use various theorems, but here i don't have any numbers, just general series. So i have problem with two of them. We know that $a_n$ is positive and $\sum a_n$ is…
Krzysztof Lewko
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Convergent sequence confusion

I am having an issue understanding why the sequence $x_n$ such that $x_{n+1} = \sqrt{2x_n +1}$, $x_1 = 1$ is convergent. After plugging numbers into the sequence we discovered that the sequence is increasing and we could not find a limit that would…
user115964
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Need confirmation of the definition of $\mathcal{L}^2$ convergence

So we say $f_n(t)$ converges in $\mathcal{L}^2$ to $f(t)$, more specifically we mean: $\lim\limits_{n\rightarrow \infty}\int_0^t|f_n(s) - f(s)|^2 ds = 0$, hence if in terms of the $\lim\limits_{n\rightarrow \infty}$ notation instead of the usual…
Yue Harriet Huang
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Finding rate of convergence

I'm trying to determine a rate of convergence for a non linear function $f(x)=x^5 + 12x^3 -130$ to find its root. Using the fixed point iteration, I am using the second form function $g(x)=(4x^5 + 36x^3 + 130)/(5x^4 + 48x^2)$. If I continue to try…
omega
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Sequence in $l^2$ that does not converge (but it should?)

Let $l^2$ be the space of all real sequences $x = (x_1,x_2, x_3,\;...)$ for which $\sum_{n=1}^\infty |x_n|^2$ converges. It can be easily verified that map $$\langle x,y\rangle = \sum_{n=1}^\infty x_ny_n$$ defines an inner product on $l^2$ and that…
Antoine
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