Is there any closed form for this expression sum_(n=0)^oo ln(n+x)

Mariah Sparks

Mariah Sparks

Answered question

2022-07-17

Infinite sum of logarithms
Is there any closed form for this expression
n = 0 ln ( n + x )

Answer & Explanation

Dominique Ferrell

Dominique Ferrell

Beginner2022-07-18Added 18 answers

It's undefined whenever x 0, and the series diverges to infinity whenever x > 0
Elianna Lawrence

Elianna Lawrence

Beginner2022-07-19Added 2 answers

Write it as n = 0 k ln ( n + x ) = ln ( n = 0 k ( n + x ) ) = ln ( Γ ( k + x + 1 ) Γ ( x ) ) . If k goes to the expression diverges, if x > 0 or x is not equal to a negative integer.

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