Calculus BC: Series
Geometric Series and the Ratio Test
A geometric series is a series of the form
ar
n
(where we take
a
and
r
to be positive). One learns in high school algebra that this series converges if and
only if
0 < r < 1
. If
ar
n
does converge, we have
ar
n = 
|
We can combine these comments about geometric series with the comparison test to yield
another test called the ratio test: given a series
a
n
with
a
n > 0
for all
n
, if
there exists a number
C
with
0 < C < 1
such that
 ≤C
|
for all
n
, then
a
n
converges.
To prove this fact, note that under the hypotheses of the theorem,
| a n≤Ca n-1≤C 2 a n-2≤ ... ≤C n-1 a 1 |
Letting
b
n = a
1
C
n-1
, so that
b
n
is a (convergent)
geometric series, we see that
a
n≤b
n
. By the comparison test,
a
n
must also converge; in fact
a
n≤
b
n = 
|
