Chapter 5.5, Problem 301E

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

301. Show that the alternating series $\cos \theta =1-\frac{{\theta }^2}{2!}+\frac{{\theta }^4}{4!}+\frac{{\theta }^6}{6!}+\mathrm{...}$ will be derived in the next chapter. Use the remainder $\left|R_N\right|\le b_{N+1}$ to find a bound for the error in estimating $\cos \theta$ by the fifth partial sum $1-{\theta }^2/2!+{\theta }^4/4!-{\theta }^6/6!+{\theta }^8/8!$ for $\theta$ =1. What $\theta =\pi /6$ , $\theta =\pi$ .