“Teach A Level Maths”
Vol. 1: AS Core Modules
33: Geometric Series
Part 2
Sum to Infinity
© Christine Crisp
Geometric series – Sum to Infinity
Module C2
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Geometric series – Sum to Infinity
Suppose we have a 2 metre length of string . . .
. . . which we cut in half
1m
1m
We leave one half alone and cut the 2nd in half again
1m
1
2
. . . and again cut the last piece in half
1m
1
2
1
2
m
m
1
4
m
m
1
4
m
Geometric series – Sum to Infinity
Continuing to cut the end piece in half, we would have
in total
1
1
2

1
4

1
8
 ...
1
2
1m
m
1
4
m
1
8
m
In theory, we could continue for ever, but the total
length would still be 2 metres, so
1
1
2

1
4

1
8
 ...2
This is an example of an infinite series.
Geometric series – Sum to Infinity
The series 1  1  1  1  . . .  2
2
4
8
is a G.P. with the common ratio r  1 .
2
Even though there are an infinite number of terms,
this series converges to 2.
Sum
Sn
Number of terms, n
Geometric series – Sum to Infinity
We will find a formula for the sum of an infinite
number of terms of a G.P. This is called “the sum to
infinity”, S

e.g. For the G.P.
1
1
2

1
4

1
8
 ...
we know that the sum of n terms is given by
1 1
a (1 r n )
Sn 
 Sn 
 
1 r
As n varies, the only part that changes is
This term gets smaller as n gets larger.
1 n
2
1  12
1 n.
2

Geometric series – Sum to Infinity
 
 12  n  0
As n approaches infinity, 1
2
We write:
As
So, for
r
1,
2

For the series
n  ,
n approaches zero.
n0
a (1 r )
Sn 

1 r
a
S 
1 r
1
1
2

1
4

1
8
1
S 
2
1
1 2
 ...
Geometric series – Sum to Infinity
However, if, for example r = 2,
r n  2n
As n increases, 2
As
n
also increases. In fact,
n  ,
2n  
The geometric series with r  2 diverges
There is no sum to infinity
Geometric series – Sum to Infinity
Convergence
If r is any value greater than 1, the series diverges.
Also, if r  1, ( e.g. r = 2 ), r n    as n  
So, again the series diverges.
If r = 1, all the terms are the same.
If r = -1, the terms have the same magnitude but
they alternate in sign. e.g. 2, -2, 2, -2, . . .
A Geometric Series converges only if the common
ratio r lies between 1 and 1.

a
S 
1 r
for
1  r  1
This can also be written as
r 1
Geometric series – Sum to Infinity
e.g. 1. For the following geometric series, write
down the value of the common ratio, r, and decide if
the series converges. If so, find the sum to infinity.
2
Solution:
r
 12
1
2

1

2
4
1
8

1
32
. . .
so r does satisfy 1 < r < 1
a
2
S 
 S 
1 r
1   14
 
 S 
The series converges to
1 6
8
5
or 1  6
Geometric series – Sum to Infinity
SUMMARY

A geometric sequence or geometric progression
(G.P.) is of the form
a, ar , ar 2 , ar 3 , . . .

The nth term of an G.P. is

The sum of n terms is
a (1 r
Sn 
1 r

n
)
or
The sum to infinity is
un  ar
n 1
a ( r n 1 )
Sn 
r 1
a
S 
;  1  r  1 or
1 r
r 1
Geometric series – Sum to Infinity
Exercises
1. For the following geometric series, write down the
value of the common ratio, r, and decide if the
series converges. If so, find the sum to infinity.
(a ) 2  3  92  27
. . .
4
(b) 3  1  13  19  . . .
Ans: (a) r   3 so the series diverges.
2
(b) r   1 so the series converges.
3
a
S 
1 r

S 
3
 
1   13
9

4
or 2  25
Geometric series – Sum to Infinity
Geometric series – Sum to Infinity
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Geometric series – Sum to Infinity
We will find a formula for the sum of an infinite
number of terms of a G.P. This is called “the sum to
infinity”, S

e.g. For the G.P.
1
1
2

1
4

1
8
 ...
we know that the sum of n terms is given by
1 1
a (1 r n )
Sn 
 Sn 
 
1 r
As n varies, the only part that changes is
1 n
2
1  12
1 n.
2

This term gets smaller as n gets larger.
As n approaches infinity, 1 n approaches zero.
2
1 n  0
We write: As n   ,

2 
Geometric series – Sum to Infinity
So, for
As
r
a (1 r n )
Sn 
1 r
1
2
n  ,
 
1
2
For the series 1 
n
1
2

0

1
4

1
8
a
S 
1 r
 ...,
S 
However, if, for example r = 2,
r n  2n
As n increases, 2
As
n
also increases. In fact,
n  ,
2n  
The geometric series with r  2 diverges
There is no sum to infinity
1
1
1
2
2
Geometric series – Sum to Infinity
Convergence
If r is any value greater than 1, the series diverges.
Also, if r  1, ( e.g. r = 2 ),
rn  
as
n
So, again the series diverges.
If r = 1, all the terms are the same.
If r = -1, the terms have the same magnitude but
they alternate in sign. e.g. 2, -2, 2, -2, . . .
A Geometric Series converges only if the common
ratio r lies between 1 and 1.

a
S 
1 r
for
1  r  1
( or r  1 )
Geometric series – Sum to Infinity
SUMMARY

A geometric sequence or geometric progression
(G.P.) is of the form
a, ar , ar 2 , ar 3 , . . .

The nth term of an G.P. is

The sum of n terms is
a (1 r
Sn 
1 r

n
)
or
The sum to infinity is
un  ar
n 1
a ( r n 1 )
Sn 
r 1
a
S 
;  1  r  1 or
1 r
r 1