I saw someone do this (in a youtube video):
∞∑n=1Γ(s)ns=Γ(s)∞∑n=11ns=Γ(s)ζ(s)=∞∑n=1{∫∞0us−1e−nu du}=
∫∞0us−1{∞∑n=1e−nu} du=∫∞0us−1⋅1eu−1 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:
limu→0us−1⋅1eu−1
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