Chapter 10.6, Problem 31E

Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − S_n| in using the nth partial sum S_n to estimate the value of the series S.

31. ${\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{{(-1)}^k}{k^4+k+1}}$; n = 5