Questions tagged [divergent-series]

Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$. Q: Suppose for some reason (e.g. numerical) we know that there is ...

Playing with identities ($1$) and ($2$) from this blog post and infinite geometric series, I've noticed the following. For $x > 1$, the following series is convergent: $$\sum_{n=0}^{\infty} e^{(2n ...

In analytic number theory, it is common to prove that $$\sum_{n\leq N} \frac{a_n}{n} = o(\log N)\tag{$\star$}\label{476699_star}$$ for some sequence $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{C}$. (It is ...

Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$ is divergent

When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients $$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...