Chapter 8.3, Problem 52PE

a. Determine the partial fraction decomposition for $\frac{3}{n\left(n+3\right)}.$

b. Use the partial fraction decomposition for $\frac{3}{n\left(n+3\right)}$ to rewrite the infinite sum $\frac{3}{1\left(4\right)}+\frac{3}{2\left(5\right)}+\frac{3}{3\left(6\right)}+\frac{3}{4\left(7\right)}+\frac{3}{5\left(8\right)}\cdot \cdot \cdot$

c. Determine the value of $\frac{1}{n+3}$ as $n\to \infty .$

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