How do I evaluate the sum sum_(k=1)^oo(ln(1+1/(k+a))-ln(1+(1)/(k+b))) where 0<a<b<1?

propappeale00

propappeale00

Answered question

2022-10-14

How do I evaluate the sum k = 1 ( ln ( 1 + 1 k + a ) ln ( 1 + 1 k + b ) )
where 0 < a < b < 1?
Hints will be appreciated
Thanks

Answer & Explanation

erkvisin7s

erkvisin7s

Beginner2022-10-15Added 12 answers

We have
k = 1 n ln ( 1 + 1 k + a ) = ln ( k = 1 n ( 1 + 1 k + a ) ) = ln ( k = 1 n k + 1 + a k + a ) = ln ( n + 1 + a 1 + a )
and thus
k = 1 n ln ( 1 + 1 k + a ) ln ( 1 + 1 k + b ) = ln ( n + 1 + a 1 + a ) ln ( n + 1 + b 1 + b ) = ln ( 1 + b 1 + a ) ln ( n + 1 + b n + 1 + a ) ln ( 1 + b 1 + a ) ,
as n , since
ln ( n + 1 + b n + 1 + a ) = ln ( 1 + b a n + 1 + a ) 0.

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