Chapter 5.5, Problem 312E

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,

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

312. [T] Plot the series ${\displaystyle \underset{n=1}{\overset{100}{\sum }}\frac{\cos \left(2\pi nx\right)}{n}}$ for $0\le x<1$ and describe its graph.