Infinite Geometric Series
Imagine the infinite ways you can top a hotdog. Relish looks kind of bland after you consider what strawberry sauce and sprinkles could do to a hotdog.
Infinite geometric series, which are just specially dressed finite geometric series, have some wacky properties that can make them interesting. The standard infinite geometric series looks like
We already know for a geometric series the nth partial sum is
where r is the ratio of consecutive terms, a is the first term, and n is the number of terms.
Use the formula for Sn to find
(a) |r| < 1
(b) 1 < |r|
(c) r = 1 (hint: look at the expanded geometric series)
(d) r = -1 (hint: look at the expanded geometric series)
(a) Using the formula for Sn,
As n approaches ∞, since |r| < 1 the quantity rn approaches 0.
(b) Again we use the formula for Sn. Since |r| > 1, the quantity rn approaches ∞ as n approaches ∞.
The numerator of the fraction will get farther from 0 without bound as n approaches ∞, so this limit doesn't exist.
(c) Following the hint, look at the expanded series. When r = 1 the geometric series looks like
Since every power of 1 is 1, this is the constant series
a + a + a + ...
The nth partial sum is Sn = a · n.
does not exist. We found this result earlier when looking at a constant series
(d) Looking at the expanded series, when r = -1 we get
which is the alternating series
a – a + a – a + a – a.
The partial sums are
S1 = a = a
S2 = a – a = 0
S3 = a – a + a = a
S4 = a – a + a – a = 0
and so on.
Since the partial sums bounce back and forth between two values,
does not exist.
The last example covered all cases for r. We weren't playing the role of Oedipus here, leading you blindly down a dangerous path. We know that the sum of an infinite series, if it exists, is the limit of the partial sums Sn. In the previous exercises we found that
only exists when |r| < 1, in which case
This means when |r| < 1, the sum of the infinite geometric series
In all other cases (when |r| ≥ 1) the limit of partial sums
doesn't exist. This means when |r| ≥ 1 the sum of the infinite geometric series
doesn't exist either.
All of this means that an infinite geometric series converges when the ratio r has magnitude strictly less than 1. An infinite geometric series diverges in all other cases.
Finding the sum of an infinite geometric series is easier than finding a partial sum, because we only need to know a and r. We don't need to worry about how many terms there are. There are infinitely many.
The sum of a convergent geometric series is
where a is the first term of the series and r is the ratio.