Series Convergence

Date: 02/28/2001 at 04:26:01
From: Julian Havil
Subject: Convergence

Please can you tell me why 1 + 1/2^z + 1/3^z + ... converges for 
Re(z) > 1?

Thank you.

Date: 02/28/2001 at 15:24:16
From: Doctor Rob
Subject: Re: Convergence

Thanks for writing to Ask Dr. Math, Julian.

Take the absolute value of each term.

   |1/n^z| = |n^(-z)|,
           = |e^(-z*ln[n])|,
           = |e^(-[Re(z)+i*Im(z)]*ln[n])|,
           = e^(-Re[z]*ln[n]),
           = n^(-Re[z]),
           = 1/n^Re(z).

Now the series SUM 1/n^a converges if and only if a > 1.  (Do you
need a proof of that, too?)  That means that the original series is
absolutely convergent if Re(z) > 1, so it is convergent.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/