Finding series is convergent or divergent and sum! (Calc. 2)

Question

Determine whether the following series is convergent or divergent . If convergent, find its sum.

Problem: sum of : 2/(n^2-1), from 2 to infinity

Answer 1

THe series is positive, continuous, and decreasing.

The integral test says that the series and the integral either both converge or both diverge.

integral is done with partial fraction decomposition

1/(n^2 - 1) = A/(n-1) + B/(n+1)

1 = A(n+1) + B(n-1)

1 = An + A + Bn - B

A+B = 1

A-B = 0

2a = 1

a=1/2 ---> b = 1/2

The integral is (-1/2) {ln |n+1| + ln|n-1| }

L'Hopitals rule for infinity - infinity says log shall tend to zero.

The limit is ln(3)/2 so it converges