How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #ln(1/2)+ln(2/3)+ln(3/4)+...+ln(n/(n+1))+...#?
We know that
So, we can rewrite the sum
Slightly rearrange the terms:
See how almost all of the terms cancel out? This is called a telescoping sum. Everything simplifies to
Now, to find the sum of the infinite series, set
So, the sum of the infinite series diverges.