Chapter 8.3, Problem 68E

Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {S_n}. Then evaluate $\underset{n\to \infty }{\lim }{S}_n$ to obtain the value of the series or state that the series diverges.

68. ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}(ta{n}^{-1}(k+1)-ta{n}^{-1}k)}$