Mathematics Stack Exchange
Mathematics Stack Exchange

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 .

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If an improper integral equals some value L, does that imply it is divergent?

I worked out $\int _1^{\infty }\frac{\sqrt{4x+5}}{x^2}\:dx$ to have an undefined limit, which tells me it diverges. I'm still trying to fully understand improper integrals so somebody correct me if this integral doesn't diverge. I wanted to ask, if…
Tingo Hugo
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What is the connection between the root test and the radius of convergence of the power series?

$$R=\dfrac{1}{\limsup_{n\to\infty}|a_n|^{\frac{1}{n}}} $$ I've been reading this paper regarding asymptotic growth, and I stumbled upon this relation between the radius of convergence and the root test. From my knowledge, the root test shows if the…
Mihailo
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Determine whether the integral diverges

To determine whether the integral diverges, you need to look at the ratio of the maximum powers of the numerator and denominator. If the ratio is greater than 1, then the integral is divergent. I apply this rule to my example, where you need to find…
Антон
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Is $\sum_{n=1}^\infty \frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}$ divergent?

How to show that $$\sum_{n=1}^\infty \bigg(\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}\bigg)$$ is divergent? I tried multiplying by the conjugate. I got $$({\sqrt{n+2}-\sqrt{n}})({\sqrt{n+1}+\sqrt{n}})$$ How to prove that it is divergent? Is…
Adrianna
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Dirichlet Test vs. Alternating series test

One can use Dirichlet's test to prove the alternating series test quite easily. I am wondering if there is a simple proof of Dirichlet's test by assuming the alternating series test holds. An assumption made in the hypothesis of Dirichlet's test is…
jenny9
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Are a_n and b_n convergent and what can be said about the limit values

Let $0 < b_1 ≤ a_1$ be given. We define for $n \in \mathbb{N}$: \begin{equation} b_{n+1}:=\sqrt{a_nb_n} \\ a_{n+1}:= \dfrac{a_n+b_n}{2} \end{equation} Are $a_n$ and $b_n$ convergent and what can be said about the limit values? I have spent hours for…
battel101
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Relationship between f divergence and KL divergence

I read that the Kullback-Leibler divergence defined as $$D_{KL} = \int_{-\infty}^{\infty}p\left(x\right)\cdot log\left(\frac{p(x)}{q(x)}\right)dx$$ can be seen as a special case of the f-divergence $$D_{f} =…
tintinnabulum
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Calculating the interval and radius of convergence

I have $$s_n=\frac{-6^n(x+8)^n}{\sqrt{n}}$$ and I have to find the interval and radius of convergence. Using ratio test: $$\frac{-6^{n+1}(x+8)^{n+1}}{\sqrt{n+1}}\cdot \frac{\sqrt{n}}{-6^n(x+8)^n}$$ $$\lim_{n\rightarrow…
Luthier415Hz
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whether it's absolutely convergent

I want to know how to use a simple method to prove $\sum\frac{\sin(n)sin(n^2)}{n}$ is absolutely convergent?
lynor
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Convergence in q-norm expectation question

Use the triangle inequality for the norm on $L^{q}$ to show that if $q \geq 1$ and $X_{n} \rightarrow^{q.m} X$ (converges in q mean) then $$E[|X_{n}|^{q}] \rightarrow E[|X|^{q}]$$ So far, I've tried figuring out how to extend the triangle inequality…
ballthrowaway
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How to test whether an infinite sum has converged up to a tolerance

I have a function F(x) that is an infinite sum that I know converges. I want to evaluate F(x) for a certain value of x and I have a tolerance. To give an example, let's say I have a formula for pi expressed as infinite series as the following: And…
Ahmed Abdullah
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proof of the convergence of a recursive sequence

The following recursive sequence seems to converge towards 8. How could you prove this? $$a_{0} = 0$$ $$a_{n+1} = 4\cdot\sqrt[4]{8+a_{n}}$$ Appreciate any responses.
Martyn Laokoon
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How can we, for each x, find an N so that the power series of the exp. function converges?

So as we know, the power series by Euler of the exponential function converges for every x. That means that for all x and for all epsilon, we can find an N so that the series converges. Is there a way to compute this N for each x? In other words,…
anon
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For which N does the limit definition by Euler of the exponential function converge?

The limit definition by Euler is given as $e^z = \lim_{n \to \infty} (1 + \frac{z}{n})^n$ and thus for large $N$ we have $e^z = (1 + \frac{z}{N})^N$. Now my question is: how "fast" does this approximation converge? I.e., how do I have to choose $N$…
anon
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Alternating series failing the first criteria

I am taking a Calculus course and pretty new here, but I cant find a solution to this problem anywhere. The series is obviously alternating, so I want to check if it meets the criteria for the alternating series test. It fails the first criteria,…
LukeDalton
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