Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series

P1^

3

Note

You may have noticed that the sum of the series 0.2 + 0.02 + 0.002 + ... is 0.2, and ̇

that this recurring decimal is indeed the same as^29.

ExamPlE 3.10 The first three terms of an infinite geometric progression are 16, 12 and 9.

(i) Write down the common ratio.

(ii) Find the sum of the terms of the progression.

SOlUTION

(i) The common ratio is^34.

(ii) The sum of the terms of an infinite geometric progression is given by:

S a

∞ r

=

1–.

In this case a = 16 and r =^34 , so:

S∞=^16 =

1

3 64

– 4

.

●?^ A paradox

Consider the following arguments.

(i) S = 1 − 2 + 4 − 8 + 16 − 32 + 64 − ...

⇒ S = 1 − 2(1 − 2 + 4 − 8 + 16 − 32 + ...)

= 1 − 2 S

⇒ 3 S = 1

⇒ S=^13.

(ii) S = 1 + (− 2 + 4) + (− 8 + 16) + (− 32 + 64) + ...

⇒ S = 1 + 2 + 8 + 32 + ...

So S diverges towards +∞.

(iii) S = (1 − 2) + (4 − 8) + (16 − 32) + ...

⇒ S = –1 − 4 − 8 − 16 ...

So S diverges towards −∞.

What is the sum of the series: 13 , +∞, −∞, or something else?