The Chemistry Maths Book, Second Edition

7.4 Infinite series 203

EXAMPLE 7.7Find the sum of the first nterms of the series 2 1·1 51 + 13 1·1 71 + 14 1·1 91 +1-

The general term isu

r

1 = 1 (r 1 + 1 1)(2r 1 + 1 3)forr 1 = 1 1, 2, 3,1=Then

u

r

1 = 12 r(r 1 + 1 1) 1 + 13 r 1 + 13

and

0 Exercise 40

7.4 Infinite series

The limit of a sequence of partial sums is the (sum of the) infinite series

(7.16)

where the dots mean that the sum is to be extended indefinitely. The series has a

sum only if the limit is finite and unique; that is, when the sequence of partial sums

converges.

The geometric series

The geometric series is the limit of the sequence of partial sums

(whenx 1 = 1 1 the sum is nand the series diverges). The sequence of sums shows all six

types of behaviour illustrated in Figure 7.1 (type (b) only for the trivial casex 1 = 10 ),

and converges only when|x| 1 < 11. Thus, when|x| 1 < 11 ,x

n

1 → 10 asn 1 → 1 ∞, and

lim

n

x

xx

n

→

−

−

















=

−

∞

1

1

1

1

Sxx x

x

x

x

n

n

n

=+ + + + =

−

−

,≠

−

1

1

1

1

21



Suuuu

n

r

n

n

==+++

→

=

























∑

lim

∞

1

123



=++

1

6

42135

2

nn n()

=× 2 + + +× + +×

1

3

123

1

2

nn n()( ) ()nn 13 n

r

n

r

r

n

r

n

r

n

urr r

== ==

∑∑ ∑∑

=+++

11 11

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