Friday, 18 November 2016

limits - Strange equality involving a geometric series and gamma and zeta function

I saw someone do this (in a youtube video):



n=1Γ(s)ns=Γ(s)n=11ns=Γ(s)ζ(s)=n=1{0us1enu du}=


0us1{n=1enu} du=0us11eu1 du



But, I can follow all the steps he did but the last integral does not converge because the geometric series only hold when the real part of u is bigger then 0, but the lower bound of the integral equals 0. So why are those two things equal?



Or can we assign a value to:




limu0us11eu1

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