Find all complex numbers such that the series

converges.

We look to apply the root test, letting

Then we have

Since this series converges if this limit is less than 1 and diverges if it is greater than 1, we have the series is absolutely convergent for

And about |z|=e^(-1/85)? In this case, I know that the root test is inconclusive, but is there a way to guarantee the convergence or divergence?

With |z|=e^(-1/85), the limit of the terms of the serie goes to e^(-3/50) as n-> infinity. Therefore, the series diverge in this case.

Nice point, Rafael. But I get e^(17n/85-17n/85)=e^0=1, so the sequence goes to 1.