Chapter 5.2, Problem 117E

Each of the following infinite series converges to the given multiple of $\pi$or $1/\pi$ .

In each case, find the minimum value of N such that the Nth partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and

give the desired approximate value. Up to place, $\pi$= 3.14 1592653589793....

117. $\left[\text{T}\right]\pi =-3+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{n{2}^{n}n{!}^2}{\left(2n\right)},\text{error}\text{<0}\text{.0001}}$