How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #3/(1^2*2^2)+5/(2^2*3^2)+...+(2n+1)/(n^2(n+1)^2)+...#?
1 Answer
The partial sum is
The series converge and the sum is
Explanation:
Let's perform a partial fraction decomposition
As the denominators are the same, we compare the numerators
Let
Let
Coefficients of
Coefficients of
So,
Now we can calculate the sums
.
.
.
So,