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# Find all z such that the series (1 + 1/(5n+1))n2 |z|17n converges

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 1. Rafael Deiga says:

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?

• Rafael Deiga says:

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.

• tom says:

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

• Anonymous says:

Hello, good day to you fellow mathematicians!

The limit is in fact . Tom you must take the limit of the first expression, before the -th root. To evaluate that limit, use the Taylor expansion of log with changed to .

Cheers, and happy problem solving!