1. “Teach A Level Maths” Vol. 1: AS Core Modules
33: Geometric Series
Part 2
Sum to Infinity
© Christine Crisp

2. Module C2
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3. Suppose we have a 2 metre length of string . . .
. . . which we cut in half
We leave one half alone and cut the 2nd in half again
. . . and again cut the last piece in half

4. Continuing to cut the end piece in half, we would have in total
In theory, we could continue for ever, but the total length would still be 2 metres, so
This is an example of an infinite series.

5. The series
is a G.P. with the common ratio
Even though there are an infinite number of terms, this series converges to 2.
Number of terms, n
Sum

6. 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”,
e.g. For the G.P.
we know that the sum of n terms is given by
As n varies, the only part that changes is
This term gets smaller as n gets larger.

7. As n approaches infinity, approaches zero.
We write:
So, for ,
For the series

8. However, if, for example r = 2,
As n increases, also increases. In fact,
The geometric series with diverges
There is no sum to infinity

9. Convergence
If r is any value greater than 1, the series diverges.
Also, if r < -1, ( e.g. r = -2 ),
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.
for
This can also be written as

10. 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.
Solution:
so r does satisfy -1 < r < 1
The series converges to

11. SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
The sum of n terms is or
The sum to infinity is or

12. 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.
Ans: (a) so the series diverges.
(b) so the series converges.

14. 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.

15. 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”,
e.g. For the G.P.
As n varies, the only part that changes is
We write:
This term gets smaller as n gets larger.
As n approaches infinity, approaches zero.
we know that the sum of n terms is given by

16. So, for
However, if, for example r = 2,
As n increases, also increases. In fact,
The geometric series with diverges
For the series
There is no sum to infinity

17. Convergence
Also, if r < -1, ( e.g. r = -2 ),
If r is any value greater than 1, the series diverges.
So, again the series diverges.
A Geometric Series converges only if the common ratio r lies between -1 and 1.
for
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, . . .
( or )

18. SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
The sum of n terms is or
The sum to infinity is