Understanding Engineering Mathematics

14.9 Infinite series

An infinite series (105
➤
) is one of the form

S=u 1 +u 2 +u 3 +···+un+···

i.e. the sum of the terms of an infinite sequence. To consider the convergence of such a
series, i.e. whether it adds up to a finite quantity, we introduce some definitions. Firstly,
we pretend that the series is finite and define thenthpartial sum of the seriesas the sum
of the firstnterms:

Sn= sum tonterms =u 1 +u 2 +u 3 +···+un

Then the infinite series can be regarded as thisnth partial sum asntends to infinity, i.e.

S= sum to infinity = lim

n→∞

Sn

IfS=limn→∞Sn=l, a definite value, then we say that the seriesconvergesto the valuel.
Otherwise the series isdivergent.
In terms of sigma notation (102

➤

) we write:

Sn=

∑n

r= 1

ur

S=

∑∞

r= 1

ur

Clearly, whether or not a series converges depends on the termsurandalsoontheir
relation to each other – i.e. onur/ur− 1. This last quantity appears in theratio testfor
convergence.
One way of testing for convergence, of course, is to investigate the limit of thenth
partial sum directly.

Problem 14.10

Investigate the convergence of the geometric series

S=aYarYar^2 Y···Yarn−^1 Y···

(105

➤

)

We looked at this in Section 3.2.10. We have

Sn=a+ar+ar^2 +···+arn−^1

=

a( 1 −rn)

1 −r

(Section 3. 2. 9 )

=

a

1 −r

−

(

a

1 −r

)

rn