Series ConvergenceDate: 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/ |
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