Understanding Engineering Mathematics

The only placenoccurs here is in thern.Now,if|r|<1, thenrn‘tends to’ zero asn
gets larger and larger, i.e.rntends to zero asntends to infinity, or in symbols,rn→0as
n→∞if|r|<1. We will look further into such ‘limits’ in Chapter 14. So asntends to
infinity the second term in expression for the sumSngoes to zero and the sum to infinity is

S∞=

a

1 −r

Note that ifr≥1 then the series will not converge to a finite quantity becausernincreases
indefinitely in value asnincreases if|r|>1, whilea/( 1 −r)does not exist at all ifr=1.

Example

Consider the series we started with:

S∞= 1 +

1

3

+

1

9

+

1

27

+···= 1 +

1

3

+

(

1

3

) 2

+

(

1

3

) 3

+···

Here the first term is 1 and the common ratio is^13 :

a= 1 ,r=^13

so

S∞=

1

1 −^13

=

3

2

Solution to review question 3.1.10

Using the result thatS∞=a/( 1 −r), for the sum of an infinite GP with

first terma=^12 and common ratior=^12 we have

S∞=

1

2

1 −^12

= 1

3.2.11 Infinite binomial series

➤

89 110➤

Another important type of infinite series is that obtained by applying the binomial theorem
with negative or fractional power. Referring back to Section 2.2.13 we can write for the
binomial expansion fornan integer

( 1 +x)n= 1 +nx+···+

n(n− 1 )

2!

x^2 +

n(n− 1 )(n− 2 )

3!

x^3

+···+

n(n− 1 )...(n−r+ 1 )

r!

xr+···

Now we have certainly not proved that this holds ifnisnot an integer. For example ifn
is a negative integer, or a fraction, then it is by no means obvious that the result will still
hold. However, it can be shown that it does hold, but there is one dramatic difference ifn
is not a positive integer. In this case the series will never terminate because only positive