Chapter 5.3, Problem 176E

[T] Find the minimum value of N such that the remainder estimate ${\displaystyle {\int }_{N+1}^{\infty }f<R_N<{\displaystyle {\int }_N^{\infty }f}}$guarantees that ${\displaystyle \underset{n=1}{\overset{N}{\sum }}{a}_n}$

estimates ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{a}_n}$. accurate to within the given error.

176. $a_n=\frac{1}{n{\ln }^{2}n},{\text{error <10}}^{\text{-3}}$