The Chemistry Maths Book, Second Edition

204 Chapter 7Sequences and series

so that the geometric series is the expansion of the function in powers of x:

(7.17)

The series diverges for all other values ofx(|x| 1 ≥ 1 1).

0 Exercises 41–45

The harmonic series

Despite appearances, this series diverges. It can be written as a sum of partial sums,

in whichs

n

contains 2

n

terms of which the last, and smallest, has value 122

n+ 1

. Each

of these sums is therefore larger than 2

n

1 × 1 (1 22

n+ 1

) 1 = 1122 ; for example,

It follows that

and the series diverges.

7

7.5 Tests of convergence

The example of the geometric series demonstrates that it is straightforward to show

that a series converges, or otherwise, if a closed formula is known for the partial sums.

S>+++++ 1

1

2

1

2

1

2

1

2



ss

12

1

3

1

4

1

4

1

4

1

5

1

6

1

7

1

8

1

8

1

8

1

8

1

8

=+ > +, =+++>+++

=+ + + + + 1

1

2

123

sss

++++++++











+

1

9

1

10

1

11

1

12

1

13

1

14

1

15

1

16



S=+ + +













++++













1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

S=+ + + + 1

1

2

1

3

1

4



1

1

11

23

−

=+ + + + , ||<

x

xx x x

1

1 −x

7

This demonstration of the divergence of the harmonic series and discussions of other infinite series are found

in Oresme’s Quaestiones super geometriam Euclidis(c.1350).