Sequences and series

P1^

3

The same method can be applied to the general geometric progression to give a

formula for its value:

Sn = a + ar + ar^2 + ... + arn–1. (^) ^1
Multiplying by the common ratio, r, gives:
rSn = ar + ar^2 + ar^3 + ... + arn. (^) ^2
Subtracting ^1 from ^2 , as before, gives:
(r − 1)Sn = – a + arn
= a(rn − 1)
so Sn = ar
r
()n
()

−

−

1

1

.

This can also be written as:

Sn arr

n

= (–(–^11 )).

Infinite geometric progressions

The progression^1 ++ 21 14 ++ 81 161 +... is geometric, with common ratio^12.

Summing the terms one by one gives 11 ,,^121134 ,, 87 11615 .

Clearly the more terms you take, the nearer the sum gets to 2. In the limit, as the

number of terms tends to infinity, the sum tends to 2.

As n → ∞, Sn → 2.

This is an example of a convergent series. The sum to infinity is a finite number.

You can see this by substituting a = 1 and r = 12 in the formula for the sum of the

series:

S

ar

n r

n

= ()

1

1

–

–

giving

=× (^21) ()–()^12
n.
The larger the number of terms, n, the smaller ()^12
n
becomes and so the nearer Sn
is to the limiting value of 2 (see figure 3.3). Notice that ()^12
n
can never be negative,
however large n becomes; so Sn can never exceed 2.
Sn
n

×()()

()

11

1

1

2

1

2

–

–