series question 3

Question

Determine the sum of the series ∑n=1 to ∞ 7/(n(n+2))

Answer 1

Use Partial Fraction decomposition:

7 / [n(n+2)] = A / n + B / (n + 2)

A(n + 2) + Bn = 7

If n = 0, then we get A = 7/2 and if n = -2, we get B = -7/2

So, 7 / [n(n+2)] = 7/(2n) - 7/[2(n+2)]

Let S_n = sum of first n terms = [7/2 - 7/6] + [7/4 - 7/8] + [7/6 - 7/10] + ... +

[7/(2n-2) - 7/(2n+2)] + [7/(2n) - 7/(2n+4)]

The sum "telescopes" (.most of the terms cancel out except for 7/2, 7/4, and a few terms at the end. The terms at the end go to zero as n goes to infinity)

Sum of series = lim_(n →_∞) S_n = 7/2 + 7/4 = 21/4 = 5.25

Answer 2

7/(n(n+2)) is not a geometric rule. The rule doesn't have a common ratio r. It isn't possible to find the sum of the infinite series.