Chapter 10.4, Problem 44E

Remainders and estimates Consider the following convergent series.

1. a. Find an upper bound for the remainder in terms of n.
2. b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.
3. c. Find lower and upper bounds (L_n and U_n, respectively) on the exact value of the series.
4. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.

42. ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}k{e}^{-k^2}}$