Chapter 5.5, Problem 308E

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…

308. [T] The series ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{\sin \left(x+\pi n\right)}{x+\pi n}}$plays an important role in signal processing. Show that E ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{\sin \left(x+\pi n\right)}{x+\pi n}}$converges whenever 0 π . (Hint: Use the formula for the sum of a sum of angles.)