Chapter 10.6, Problem 71E

A diverging alternating series Consider the alternating series

${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{(-1)}^{k+1}{a}_k}=\frac{1}{1^2}-\frac{1}{1}+\frac{1}{2^2}-\frac{1}{2}+\frac{1}{3^2}-\frac{1}{3}+\cdot \cdot \cdot .$

a. Show that the individual terms of the series converge to 0. (Hint: See Exercise 70.)

b. Find a formula for S_2n. the sum of the first 2n terms of the series.

  c. Explain why the alternating series diverges even though individual terms of the series converge to 0. Explain why this result does not contradict the Alternating Series Test.