Chapter 8.3, Problem 51PE

a. Determine the partial fraction decomposition for $\frac{2}{n\left(n+2\right)}$ .

b. Use the partial fraction decomposition for $\frac{2}{n\left(n+2\right)}$ to rewrite the infinite sum $\frac{2}{1\left(3\right)}+\frac{2}{2\left(4\right)}+\frac{2}{3\left(5\right)}+\frac{2}{4\left(6\right)}+\frac{2}{5\left(7\right)}\cdot \cdot \cdot$

c. Determine the value of $\frac{1}{n+2}$ as $n\to \infty .$

d. Find the value of the sum from part (b).