Root and Ratio tests, Absolute Convergence
Last time we saw that it is much easier for an alternating series to converge
than it is for a series of positive terms. The alternating series test tells us that if
a
n
is positive and decreasing and approaches zero then
∑
(

1)
n
a
n
is convergent.
It is often important to be able to decide if a series converges because of
“cancellation” between positive and negative terms. This leads to the following
concept:
Deﬁnition: Absolute Convergence
A series
X
a
n
is absolutely convergent if the series
X

a
n

is also convergent.
A series which is not absolutely convergent (in other words a series for which
X
a
n
converges but for which
X

a
n

diverges is said to be be “conditionally convergent”.
Example 1a:
X
(

1)
n
n
2
(
n
+ 1)
3
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentExample 1b:
X
(

1)
n
n
3
We are going to learn two new convergence tests: the ratio test and the root
test. Let me ﬁrst state the two tests:
Theorem: Ratio Test
Consider the series
X
a
n
and assume that the limit of the ratio exists:
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Bronski
 Math, Calculus, Mathematical Series, lim, 1 K, 1 2K

Click to edit the document details