James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.7 Strategy for Testing Series 745
4. If you can see at a glance that lim n l ` a n ± 0, then the Test for Divergence
should be used.
5. If the series is of the form o s21d n21 b n or o s21d n b n , then the Alternating Series
Test is an obvious possibility.
6. Series that involve factorials or other products (including a constant raised to the
nth power) are often conveniently tested using the Ratio Test. Bear in mind that
| a n11ya n | l 1 as n l ` for all p-series and therefore all rational or algebraic
functions of n. Thus the Ratio Test should not be used for such series.
7. If a n is of the form sb n d n , then the Root Test may be useful.
8. If a n − f snd, where y`
f sxd dx is easily evaluated, then the Integral Test is effective
(assuming the hypotheses of this test are
1
satisfied).
In the following examples we don’t work out all the details but simply indicate which
tests should be used.
Example 1 ò
n−1
n 2 1
2n 1 1
Since a n l 1 2 ± 0 as n l `, we should use the Test for Divergence.
n
Example 2 ò
n−1
sn 3 1 1
3n 3 1 4n 2 1 2
Since a n is an algebraic function of n, we compare the given series with a p-series. The
comparison series for the Limit Comparison Test is o b n , where
b n − sn3
3n 3
− n3y2
3n 3 − 1
3n 3y2
n
Example 3 ò ne 2n2
n−1
Since the integral y`
1 xe2x2 dx is easily evaluated, we use the Integral Test. The Ratio
Test also works.
n
Example 4 ò s21d n
n−1
n 3
n 4 1 1
Since the series is alternating, we use the Alternating Series Test.
n
2
Example 5 ò
k
k−1 k!
Since the series involves k!, we use the Ratio Test.
n
Example 6 ò
n−1
1
2 1 3 n
Since the series is closely related to the geometric series o 1y3 n , we use the Comparison
Test.
n
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