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

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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

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.

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

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.

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

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

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

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

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

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.

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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

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

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 )

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