   Chapter 11.2, Problem 91E

Chapter
Section
Textbook Problem

# Consider the series ∑ n = 1 ∞ n / ( n + 1 ) ! . (a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn. (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.

(a)

To determine

To find: The partial sum of the series and guess a formula for {sn}

Explanation

Given:

The series is n=1n(n+1)!.

Calculation:

Consider the nth partial sum of the given series is n=nn(n+1)!.

Obtain the sum of s1

s1=1(1+1)!=12!=12

Thus, the first partial sum is, s1=12.

Obtain the partial sum of s2.

s2=1(1+1)!+2(2+1)!=12!+23!=12+13=3+26

That is, s2=56.

Thus, the second partial sum is, s2=(2+1)!1(2+1)!.

Obtain the partial sum of s3.

s3=1(1+1)!+2(2+1)!+3(3+1)!=12+13+324=12+8+324s3=2324

That is,

s

(b)

To determine

To prove: The formula for sn is (n+1)!1(n+1)!.

(c)

To determine

To show: The infinite series n=n(n+1)! is convergent and to find the sum of the series.

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