Series involving a Logarithm sum_(n=1)^oo [ (n)/(a)ln (1+(a)/(n))-1+(a)/(2n)]. Is there a closed form solution to this series and what is the value when a=1,2 ?

eukrasicx

eukrasicx

Answered question

2022-09-30

Series involving a Logarithm
Consider the series
n = 1 [ n a ln ( 1 + a n ) 1 + a 2 n ] .
Is there a closed form solution to this series and what is the value when a=1,2 ?

Answer & Explanation

Leslie Braun

Leslie Braun

Beginner2022-10-01Added 7 answers

For small | a | , ln ( 1 + a / n ) = j = 1 ( 1 ) j 1 ( a / n ) j / j, so your series is
n = 1 j = 2 ( 1 ) j a j ( j + 1 ) n j = j = 2 ( a ) j j + 1 ζ ( j )
I don't think this has a closed form.

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