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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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Convergence of a sequence of functions: pointwise, uniform, and almost everywhere

I'm trying to reason things out when it comes to convergence of functions and I haven't found an answer that confirms or dismisses these two points: A sequence of functions $\{f_n\}$ could converge a.e. to a function $f$, either pointwise or…
ToniAz
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Prove convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$

Taken from Soo T. Tan's Calculus textbook Chapter 9.7 Exercise 27- Define $$a_n=\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$$ One needs to prove the convergence or divergence of the series $$\sum_{n=1}^{\infty}…
idanp
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Show that the following scheme has error of second order

$$X_{n+1}=\frac{1}{2}(X_n)(1+\frac{a}{(X_n)^2})$$ From this scheme, we know by Newton-Raphson that this is for $x = \sqrt{a}$. Let's say $\theta$ be the solution. $\theta^2 = a$ To prove convergence of order p, we need to show : $|X_{n+1} - \theta|…
user1611542
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Does $(\vec{a_k})_{k\in \mathbb{N}}$ converge? And if so, what is the limit?

If $\vec{a}$ is defined like this: $$\vec{a_k}=\binom{\frac{1}{k+1}\cos{k}}{\;\;(k+1)\sin{(\frac{1}{k+1})}\;\;}$$ Does it then converge? And if so, what is the limit? Any help on how to approach this problem would be very helpful.
Void
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Convergence/Divergence of Integral, can P-test be used here?

I have an integral like this: How do I check its convergence? As far as I know, P-test can be used for integrals from 0 to 1, or A to infinity, what would I do in this case?
user349328
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Convergence of $\int_0^1 x^p ln^q \left(\frac{\ 1}{x}\right)$

So far, I determined that the integral converges for every $q>p+1$. I noticed that for example for the values $p=5, q=3$ the integral still converges. There are some values for which the integral diverges, too. I have also tried to apply the…
Yonatan Izutskiver
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Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$ I applied nth term test and was inconclusive. I tried ratio test but I don't know how to evaluate the limit. I think it is 1 therefore also inconclusive. Anyone have any ideas?
user2250537
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Is $n^{1/10}$ a cauchy sequence?

So, I stuck at a step to prove that $n^{1/10}$ is not a cauchy sequence. So, $$|y_n - y_m| = |n^{1/10} - m^{1/10}|$$ So, now how is the next step to show, that it is not a cauchy sequence?
Maxim
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Is there convergence in this sequence?

I have the sequence $a_{n} = \frac{1}{5n^2 + \cos(n\pi )+1}$ $n\in \mathbb{N}$ It's obvious that it converges to 0 but I have problems to proof it: Let $\varepsilon$ be optional and choose $N$ as $N\gt?$ Then apply to all $a_{n}$ with $n\ge…
Erdbeer0815
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Convergence of $(-1)^n/\sqrt{n}$

So, I don't know how correctly show that, $c(n) := \left(\frac{(-1)^n}{\sqrt{n}}\right)\to 0\qquad\text{ as}\quad n\to\infty.$ Should I do this with limes or by $\forall \epsilon > 0, \exists N(\epsilon)$ such that $\forall n >= N(\epsilon) :|…
Maxim
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Radius of Convergence for a complex sin function

I encountered the following power series, and while I know a couple of ways to determine radius of convergence, I wasn't able to figure out how to evaluate the appropriate limit to get said radius. Can anyone…
Marko Svraka
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Absolute-Normal convergence of a sum

I have a sum which is normal convergent and I know the limit-function of the sum, does it implies that the sum of the absolute values of each term converges to the absolute value of the function?
guest
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An examination of rates of convergence of the series for $\pi$

I got this topic for project "An examination of rates of convergence of the series for $\pi$". My question is which relative formulas or math knowledge I should research?
user326752
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Find the value of sum_(i=0)^infinity (a^i (2 b-1) (b/(1-b))^i)/(b-1)

For an application I'm working on, I need to know the value of the following expression: $\sum_{i=0}^\infty$ $a^i (2 b-1)\frac{(\frac{b}{(1-b)})^i}{(b-1)} $ (sorry it's not texed). According to WolframAlpha, it converges for |a| < |1/b - 1|, but WA…
user321374
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convergence of $f(x+y)$ to $f(x)$ for $y >$ to $0$ in $L^1$.

I have to show the following: for $f \in L^1(S^1)$ Show that the map $f(x) \to f_y = f(x+y)$ is continuous in the distance of $L^1(S^1)$. i.e. $\lim_{y \to 0} ||f_y-f||_1 = 0$ I am supposed to use that $||f_y-f||_1 \le ||f_y-f||_\infty$ and that…
lukas
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