## Geometric Series

A geometric series is such that each term is multiplied by a fixed number to get the next term.

1, 2, 4, 8, 16...

is a geometric series because each term is multiplied by a number called the common ratio – in this case 2, to get the next term. We may write

We can find a closed form expression for the n ^{th }term. If the first term isand the common ratio isthe second term will beand the third term will bethe fourth termIn general the nth term will beWe may also find an expression for the sum of a series up to n terms:

(1)

(2)

(1)-(2) givessince all the other terms canel.

We can factorise both sides to give

Ifthen we may sum an infinite number of terms and obtain a proper answer, since in the expression forabove,forHencefor

The formulae above may be used in the following ways:

The 1 ^{st } term of a geometric series is 4 and the 4 ^{th } term is 0.0625. Find

a) The sum of the series to infinity.

b)The least value of n such that the difference betweenandis less than

a) 1 ^{st } term is

4 ^{th } term is

b)

Subto givesinceis a whole number.