Strategy for Testing Series
1. If the series is of the form
X
1
n
p
, it is a
p
series
, which we know to
be convergent if
p >
1 and divergent if
p
≤
1.
2. If the series has the form
X
ar
n

1
or
X
ar
n
, it is a
geometric series
,
which converges if

r

<
1 and diverges if

r
 ≥
1. Some preliminary
algebraic manipulation may be required to bring the series into this
form.
3. If the series has a form that is similar to a
p
series or a geometric
series, then one of the
comparison tests
(Theorems 10, 11) should
be considered. In particular, if
a
n
is a rational function or an algebraic
function of
n
(involving roots of polynomials), then the series should be
compared with a
p
series (The value of
p
should be chosen by keeping
only the highest powers of
n
in the numerator and denominator). The
comparison tests apply only to series with positive terms. If
X
a
n
has some negative terms, then we can apply the Comparison Test to
X

a
n

and test for
absolute convergence
.
4. If you can see at a glance that lim
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 Spring '08
 MILTON
 Calculus, Geometric Series, Mathematical Series

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