8.4 Geometric Sequences and Series
Consider the following three sequences:
2, 2 / 3, 2 / 9, 2 / 27, 2 / 81, . . .
1, 1 / 2, 1 / 4, 1 / 8, 1 / 16, . . .
3, -1, 1 / 3, -1 / 9, 1 / 27, . . .
What each of these sequences has in common is that ratio of two successive terms is the same constant for each particular series. That constant is called the common ratio of the sequence. For the first sequence, the common ratio is one-third, for the second it is one half and for the third, it is negative one-third. Geometric series can be defined recursively in terms of their initial term and the common ratio, r, as follows:
The formula for the nth term in terms of the first term and the common ratio is
Exercise 8.4.1
Write the formula for the nth term of each of the three geometric sequences above.
A geometric series is a sum of the terms of a geometric sequence.
Suppose we wanted to add up the first 50 terms of the first geometric sequence above.
Let S denote the sum.
Then
If we multiple both sides by the common difference, we get
If we subtract the second equation from the first, all the terms subtract away except for the first term of the first equation and the last term of the second equation, giving the result that
Multiplying both sides by 3 / 2 yields
Exercise 8.4.2
Find the sums of the first ten terms of the second and third geometric series at the beginning of this section.
These results may be summarized in a formula for the sum of the first n terms of a geometric series:
where a is the first term and r is the common ratio.
If the absolute value of r is smaller than one, then rn becomes vanishingly small as n
becomes large. Thus, the sum of the infinite geometric series converges when | r | < 1.
.
Exercise 8.4.3
Find the sum of the infinite geometric series:
