   Chapter 11.2, Problem 91E

Chapter
Section
Textbook Problem

Consider the series ∑ n = 1 ∞ 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

To determine

(a)

To find:

A) The partial sum of s1, s2, s3 and s4 and

B) Guess the formula

Explanation

1) Concept:

Use the definition

2) Definition:

Given a series n=1an=a1+a2+a3+ . , then sn denotes its nth partial sum:

sn=i=1nai=a1+a2+ ..+an

3) Calculation:

A) For,

an=n=1nn+1!

s1=a1=11+1!=12!=12

s2=a1+a2=12+22+1!=12+23!=12+26=56

s3=a1+a2+a</

To determine

(b)

To prove:

The guess for the formula of sn by using mathematical induction

To determine

(c)

To show:

A) The given infinite series is convergent

B) Find the sum

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