Problem 91E

91. Consider the series $\sum_{n=1}^{\infty} n /(n+1) !$.

(a) Find the partial sums $s_{1}, s_{2}, s_{3}$, and $s_{4} .$ Do you recognize the denominators? Use the pattern to guess a formula for $s_{n} .$

(b) Use mathematical induction to prove your guess.

(c) Show that the given infinite series is convergent, and find its sum.

Step-by-Step Solution