Understanding Engineering Mathematics

So, for the series

1

2

+

1

22

+

1

23

+

1

24

+

1

25

we havea=^12 andn=5, and therefore the sum is

S 5 =

1

2

(

1 −

(

1

2

) 5 )

1 −

1

2

= 1 −

(

1

2

) 5

= 1 −

1

32

=

31

32

3.2.10 Infinite series

➤

89 110➤

Merely the mention of the term ‘infinite’ sets alarm bells ringing with many of us. An
infinite series is one whose terms continue forever, or indefinitely. We tend to indicate this
by such notation as

S∞= 1 +

1

3

+

1

9

+

1

27

+···

where the dots (technically called anellipsis– a set of three dots indicating an omission
of something understood) infer that the terms continue forever following the same pattern,

which it is hoped you have spotted. In this case we see that thenth termis

( 1

3

)n− 1
.
Of course, if we are adding up an infinite number of terms we might expect that the sum
would yield an infinite amount (i.e. a number as large as we please). This would certainly
be the case for 1+ 1 + 1 +···for example. Maybe if the terms get smaller and smaller
this would not be so? Maybe, for example, the series

1 +^12 +^13 +^14 +···

‘converges’ to a finite total? After all, the terms seem to fade away to zero ‘eventually’.
In fact the answer is no – it in fact adds up to an infinite amount, as we shall see in
Chapter 14, by one of the prettiest proofs of elementary mathematics. In fact the question
of the ‘convergence’ of such infinite series is a very difficult topic belonging to an area
of mathematics calledanalysis. We will leave such questions until Chapter 14. For the
moment we will take the pragmatic view that if we can find an expression for the sum of
a series by fair means then that series converges and we can use the sum with safety.
In particular, we cansum a GP to infinityprovided that the common ratio,r, satisfies
|r|<1. We have for the infinite GP:

S∞=a+ar+ar^2 +ar^3 +···=

∑∞

n= 1

arn−^1

The sum of the firstnterms of this is, from Section 3.2.9

Sn=

a( 1 −rn)

1 −r

=

a

1 −r

−

arn

1 −r