Question

How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given `ln(4/3)+ln(9/8)+ln(16/15)+...+ln(n^2/(n^2-1))+...`?

Answer 1

The series is `sum_{n=2}^infty ln(n^2/{n^2-1})`
The `N`th partial sum by using the properties of the logarithm is `ln(prod_{n=2}^N n^2/{n^2-1})=ln({prod_{n=2}^N n^2}/{prod_{n=2}^N(n+1)(n-1)})=ln({(N!)^2/4}/{{(N+1)N!}/4{N!}/(4N)})=ln({4N}/{N+1})`
Hence # sum ln(n^2/{n^2-1})=lim_{N rightarrow infty} ln({4N}/{N+1})=2ln(2) #