   Chapter 11.2, Problem 41E

Chapter
Section
Textbook Problem

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

To determine

The series is convergent or divergent and if it is convergent, find its sum.

Explanation

1) Concept:

i) Geometric series:

n=1arn-1=a+ar+ar2+

is convergent, if r<1 and its sum is

n=1arn-1=a1-r  r<1

If r1, the geometric series is divergent

ii) Theorem: If n=1an & n=1bn  are convergent series, then the series  n=1(an+bn) &

n=1(an+bn)=n=1an+n=1bn

iii) A series n=1an is convergent if the sequence {sn} is convergent and limnsn=s exists as a real number and the sum of the series is the limit of the sequence of the partial sums {sn}n=1an=limn sn

2) Given:

n=11en+1nn+1

3) Calculation:

Consider the given series.

n=11en+1nn+1

By using the concept,

=n=11en+n=11nn+1

Consider  n=11en

n=11en=n=11e·1en-1

n=11e·1en-1  is a geometric series.

Comparing the series n=11e·1en-1 withn=1a·rn-1 a=1e  &  r=1e

r= 1e 0.37<1

Therefore, the series n=11en  is convergent.

Now find the sum.

n=11e·1en-1=1e1-1e=1e-1

n=11en=1e-1

Now, consider   n=11nn+1

To check the convergence of series, use the definition of a convergent series and compute the partial sum

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