How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1/(1*2)+1/(2*3)+...+1/(n(n+1))+...#?
Given the series:
we can determine that is convergent using the direct comparison test since:
and the series:
is convergent based on the
Decompose now the general term using partial fractions:
and note that in the partial sum:
all terms cancel each other except the first and the last, so:
is the sum of the series.
The answer is