# Questions tagged [tauberian-theory]

Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result.

33 questions
Filter by
Sorted by
Tagged with
1 vote
54 views

44 views

### Negativity of an analytic function on $[0,1)$

I want to show the following function is negative for $z\in [0,1)$: $$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}.$$ By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. I ...
• 389
83 views

• 413
786 views

### Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit $$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$ It is clear that the ...
• 935
1 vote
43 views

89 views

### Limiting behavior of a function defined by a Lambert-type series

Fix a positive real constant $\omega$, and let $\left\{ c_{n}\right\} _{n\geq1}$ be a sequence of real numbers so that the series: $$\sum_{n=1}^{\infty}\frac{c_{n}}{n^{\omega}}$$ converges ...
• 2,219
98 views

• 2,219