The sum to infinity of a geometric series can be found using a formula if the series is convergent.
Calculating the sum to infinity
In a convergent geometric series, as n approaches infinity, the sum Sn will approach a finite value. This means that a sum can be calculated. A geometric series is convergent when ∣r∣<1 where r is its common ratio.
S∞=1−ra
Deriving the formula
The formula for a geometric series for a finite value of n:
Sn=1−ra(1−rn)
A geometric series is convergent when ∣r∣<1. Considering this as n becomes larger towards:
rn→0
This means when calculating the sum to infinity:
S∞=1−ra(1−rn)→1−ra(1−0)→1−ra
Example 1
A geometric sequence has a third term of 0.25 and sixth term of 0.03125. First show that this series is convergent, then calculate S∞.
Use the nth term to form simultaneous equations:
arn−1arn−1=ar2=0.25=ar5=0.03125
Solve for r:
ar2ar5=r3=0.250.03125
r3=0.125
r=30.125=0.5
The series is convergent because r<1.
Solve for a:
a(0.5)2=0.25
a=0.520.25=1
Find S∞:
S∞=1−ra=1−0.51=0.51
S∞=2
Note:Each term in this sequence is the previous term halved, starting from 1. Adding ad infinitum gives 2.