The Chemistry Maths Book, Second Edition

208 Chapter 7Sequences and series

Alternating series

If the terms of the seriesa

1

1 + 1 a

2

1 + 1 a

3

1 +1-become progressively smaller and alternate

in sign then the series converges. For example, the alternating harmonic series

converges. We will see in Section 7.6 that the sum of this series is ln 1 2.

7.6 MacLaurin and Taylor series

Power series

A power series in the variable xhas the form of an ‘infinite polynomial’

c

0

1 + 1 c

1

x 1 + 1 c

2

x

2

1 + 1 c

3

x

3

1 +1-

wherec

0

, c

1

, c

2

,1=are constants. The convergence properties of such series can be

investigated by the methods described in the previous section. Thus, applying the

ratio test, a power series converges when

or, equivalently, when

(7.18)

where Ris called the radius of convergenceof the series. The series is therefore

convergent when|x| 1 < 1 R; it diverges when|x| 1 > 1 R, and the casex 1 = 1 ±Rhas to be

tested by other methods.

The geometric and exponential series are examples of power series. The

geometric series has radius of convergenceR 1 = 11 , the exponential series hasR 1 = 1 ∞

(see Examples 7.9).

EXAMPLES 7.11Radius of convergence

(i) The coefficient of x

n

in the series

is. By the ratio test,|c

n

2 c

n+ 1

| 1 = 1 (n 1 + 1 1) 2 n 1 → 11 asn 1 → 1 ∞and the radius of

convergence isR 1 = 11. The series therefore converges when|x| 1 < 11 and diverges when

|x| 1 > 11. It also diverges when x 1 = 11 , when it is the harmonic series, but converges

to ln 1 2 whenx 1 = 1 − 1 (see the MacLaurin series for the logarithmic function).

c

n

n

=

1

x

n

n

n=

∑

1

∞

||< =

→

+

x

c

c

R

n

n

n

lim

∞

1

lim lim

nn

cx

cx

x

c

c

n

n

n

n

n

n

→→

+

+

+

=| | <

∞∞

1

1

1

1

1

1

2

1

3

1

4

−+−+