There is given a sequence $$\{a_{n}\}_{n>0}$$ of real numbers with following conditions:
- $$\sum_{n=1}^{\infty}a_{n}>0 ;$$
- $$\sum_{n=1}^{\infty}a_{n}\log n=0 ;$$
- $$\sum_{n=1}^{\infty}a_{n}(\log n)^{2}<0 .$$
What could be written about that sequence? I know, that i must be specific, but i am looking for properties of sequence or partional sum such as
- asymptotic growth;
- distribution among real numbers;
- intervals of monotonicity;
- modality;
- some specific behaviour of subsequences;
- etc.
I will be grateful for any answer.
edit: What if we use this substitution : $$\ a_{n}=\frac{c_{n}}{n^{x}}$$ Take a look at the sequence: $$\{c_{n}\}_{n>0}$$ How does this new defined sequence behave? It determines Dirichlet series with real domain. I am curious when this series is somewhere concave and how it relate to its coefficients.
Sums impose only one constraint on a series and you have an infinite number of degrees of freedom, so there is not much that can be said. To make the last sum converge we need the $a_n$ to go toward zero fast enough. To get the last negative and the second zero we need many of the terms far out to be negative. Only the first sum includes a contribution from $a_1$, so we just have to make $a_1$ large enough to make the first sum positive.