Chapter 5.2, Problem 120E

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....

120. $\frac{1}{12\pi }={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^k\left(6k\right)!\left(13591409+545140134k\right)}{\left(3k\right)!{\left(k!\right)}^3{640320}^{3k+3/2}}}\text{error}{\text{<10}}^{\text{-5}}$