   Chapter 11.2, Problem 41E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent. If it is convergent, find its sum.41. ∑ n = 1 ∞ ( 1 e n + 1 n ( n + 1 ) )

To determine

Whether the series is convergent or divergent and obtain the sum if the series is convergent.

Explanation

Given:

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

Result used:

(1) The geometric series n=1arn1 (or) a+ar+ar2+ is convergent if |r|<1 and its sum is a1r, where a is the first term and r is the common ratio of the series.

(2) If limnsn=L, then n=1an=L.

Calculation:

The given series can be written as follows,

n=1(1en+1n(n+1))=n=11en+n=11n(n+1)

=n=1(1e)n+n=11n(n+1) (1)

Consider the series, n=1(1e)n=(1e)+(1e)2+(1e)3+.

Clearly, it is geometric series with the first term of the series, a=1e and the common ratio of the series is,

r=(1e)21e=1e

The absolute value of r is,

|r|=|1e|=1e<1

Since |r|<1 and by the Result (1) stated above, the series is convergent.

Thus, the series n=1(1e)n is convergent.

Obtain the sum of the series.

n=1(1e)n=1e11e=1ee1e=1eee1=1e1

Therefore, the sum of the series n=1(1e)n=1e1. (2)

Consider the series, n=11n(n+1).

Obtain the partial fraction of 1n(n+1).

1n(n+1)=a0n+a1n+1 (3)

Multiply the equation by n(n+1),

n(n+1)n(n+1)=a0n(n+1)n+a1n(n+1)n+11=a0(n+1)+a1n1=(a0+a1)n+a0

Equate the coefficient of n and the constant term on both sides,

a0+a1=0

a0=1

Solve the above equations and obtain a0=1 and a1=1

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 