How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1/(1*3)+1/(2*4)+1/(3*5)+...+1/(n(n+2))+...#?
2 Answers
The partial sum is
The series converge to
Explanation:
Let's perform a decomposition into partial fractions
We compare the numerators
When
When
Therefore,
Partial sum
The series converge to
Explanation:
We can determine that the series is convergent by direct comparison, since for
and:
is convergent.
To determine the sum we can write the general term of the series as:
so the the partial sums are:
and we can see that in every partial sum, all the intermediate terms cancel two by two except for:
so that the sum of the series is: