Chapter 5.5, Problem 306E

The following alternating series converge to given multiples of $\pi$ . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in

each case. Up to 15 decimals places,

$\pi$ =3.141592653589793…

306. $\left[\text{T}\right]\frac{\pi }{4}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{2n+1},\text{error}\text{<0}\text{.0001}}$