Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

If $\sum_n a_n$ is convergent, then is $\sum_n a_n^3$ convergent?

If $\sum_n a_n$ is convergent, then is $\sum_n |a_{n+1}^2-a_n^2|$ convergent?

If $\sum_n a_n$ is convergent, then is $\sum_n a_n^2 e^{int}$ convergent?

If $a_n=\frac{j^n}{n^{1/3}}$, then $\sum_n a_n$ is convergent, but $\sum_n a_n^3=\sum_n \frac{1}{n}=+\infty$.

If $a_n = \frac{i^n}{\sqrt{n}}$, then $\sum_n a_n$ is convergent, but $\sum_n |a_{n+1}^2-a_n^2|\geq \sum_n \frac{1}{n}=+\infty$.

If $a_n =\frac{e^{-int/2}}{\sqrt{n}}$, then $\sum_n a_n$ is convergent, but $\sum_n a_n^2 e^{int}=\sum_n \frac{1}{n}=+\infty$.

user contributions licensed under cc by-sa 3.0 with attribution required