Notes on the Integral Test
The Integral Test is a method for determining the convergence or divergence of a
series of positive terms. It has the advantage that, unlike some of the other tests we will
discuss, it always provides an answer to the question of convergence. The drawbacks are:
1) there are a number of conditions that must be satisfied in order to apply the test;
2) one must be able to evaluate an improper integral, which may be difficult to
accomplish for a given series.
The Integral Test
: Let
n
a
∑
denote a given series for which the terms of the series are
eventually all positive. Suppose that
f
is a function for which
(
29
n
f
n
a
=
for all large
values of
n
. Furthermore suppose that f is positive, continuous, and decreasing on some
interval
[
29
,
M
∞
. Then
n
a
∑
converges if and only if the improper integral
(
29
M
f
x dx
∞
∫
converges.
There are several observations that one must keep in mind when applying the
Integral Test.
A.
While it does not matter about the first few terms of the series, eventually all of
the terms
n
a
must be nonnegative. Most of the time the series will be a series of
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 Spring '11
 BUEKMAN
 Calculus, Mathematical analysis

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