Divergence of $\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}....$

Question

Assume that we know this converges. $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+....$$

Is it possible to detect for which largest $0<s<1$ the sum below diverges?

$$\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}....$$

I was thinking about Riemann zeta function and i came up with this question. As you know for $t>1$, $\zeta (t)$ is convergent but $\zeta (1)$ is divergent.

$a_{1},a_{2},....,s$ are positive real numbers.

Sorry for my terrible English!

Answer 1

Take the convergent series $$ \sum_{n=2}^\infty \frac 1{n(\log n)^2}, $$ however $$ \sum_{n=2}^\infty \frac 1{n^s(\log n)^{2s}}, $$ is divergent for all $s\in (0,1)$.