 # Is there any closed form for this expression sum_(n=0)^oo ln(n+x) Mariah Sparks 2022-07-17 Answered
Infinite sum of logarithms
Is there any closed form for this expression
$\sum _{n=0}^{\mathrm{\infty }}\mathrm{ln}\left(n+x\right)$
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It's undefined whenever $x\le 0$, and the series diverges to infinity whenever $x>0$

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Write it as $\sum _{n=0}^{k}\mathrm{ln}\left(n+x\right)=\mathrm{ln}\left(\prod _{n=0}^{k}\left(n+x\right)\right)=\mathrm{ln}\left(\frac{\mathrm{\Gamma }\left(k+x+1\right)}{\mathrm{\Gamma }\left(x\right)}\right)$. If $k$ goes to $\mathrm{\infty }$ the expression diverges, if $x>0$ or $x$ is not equal to a negative integer.

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