   Chapter 11.2, Problem 77E

Chapter
Section
Textbook Problem

In Example 9 we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that e x > 1 + x for any x > 0 . (See Exercise 6.2.109.)If s n is the nth partial sum of the harmonic series, show that e S n > n + 1 . Why does this imply that the harmonic series is divergent?

To determine

To show:

That eSn>n+1 if sn is the nth  partial sum of the harmonic series.

Explain why this implies that the harmonic series is divergent.

Explanation

1) Concept:

Use the partial sum of a harmonic series and the given information to get the required result. Then, use the definition of the sum of an infinite series to explain that the harmonic series is divergent.

i) Harmonic series:

Harmonic series is n=11n=1+12+13+

The partial sum is Sn=1+12+13++1n

ii) Definition:

Let Sn be the sequence of partial sums for the series n=1an.

If the sequence Sn is convergent and limnSn=s  exists as a real number, then the series n=1an is convergent and n=1an=s

If the sequence Sn is divergent, then the series n=1an is divergent.

2) Given:

ex>1+x  for any  x>0

3) Calculation:

Sn is a partial sum of the harmonic series

Sn=1+12+13++1n

eSn=e 1 + 12 + 13 ++ 1n=e1e1/2e1/3 e</

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