Chapter 5.5, Problem 311E

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,

   $\pi =\mathrm{3.141592653589793...}$

31. $\left[\text{T}\right]\text{Plot}\text{the series }{\displaystyle \underset{n=1}{\overset{100}{\sum }}\frac{\sin \left(2\pi nx\right)}{n}}$ for 0 $\le$x<1