STRATEGY FOR TESTING SERIES We now have several ways of testing a series for convergence or divergence; the problem
11.7 STRATEGY FOR TESTING SERIES We now have several ways of testing a series for convergence or divergence; the problem
is to decide which test to use on which series. In this respect, testing series is similar to integrating functions. Again there are no hard and fast rules about which test to apply to a given series, but you may find the following advice of some use.
It is not wise to apply a list of the tests in a specific order until one finally works. That would be a waste of time and effort. Instead, as with integration, the main strategy is to classify the series according to its form.
1. If the series is of the form 1 冘 p 兾n , it is a -series, which we know to be convergent p
if and p 1 divergent if . 1
2. If the series has the form
or
冘 n ar 1 冘 ar , it is a geometric series, which converges if ⱍ r ⱍ and diverges if ⱍ r ⱍ 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 -series or a geometric series, then p
one of the comparison tests should be considered. In particular, if a n is a rational
function or an algebraic function of (involving roots of polynomials), then the n series should be compared with a -series. Notice that most of the series in Exer- p cises 11.4 have this form. (The value of should be chosen as in Section 11.4 by p keeping only the highest powers of in the numerator and denominator.) The com- n
parison tests apply only to series with positive terms, but if 冘 a n has some negative terms, then we can apply the Comparison Test to 冘 ⱍ a n ⱍ and test for absolute
convergence.
4. If you can see at a glance that lim
a n 苷 0 , then the Test for Divergence should
be used.
5. If the series is of the form 1 b n or 冘 n 冘 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 n th power) are often conveniently tested using the Ratio Test. Bear in mind that
for all -series and therefore all rational or algebraic p functions of . Thus the Ratio Test should not be used for such series. n
a 1 ⱍ l 兾a n ⱍ 1 as
7. If a n is of the form 共b n n 兲 , then the Root Test may be useful.
8. If a n 苷 f 共n兲 , where x 1 f 共x兲 dx is easily evaluated, then the Integral Test is effective
(assuming the hypotheses of this test are satisfied). In the following examples we don’t work out all the details but simply indicate which
tests should be used.
V EXAMPLE 1 1
n苷 兺 1
Since a n l 1 2 苷 0 as , we should use the Test for Divergence.
sn 3 1
EXAMPLE 2
n苷 兺 1 3n 3 4n 2 2
Since a n is an algebraic function of , we compare the given series with a -series. The n p
CHAPTER 11 INFINITE SEQUENCES AND SERIES
comparison series for the Limit Comparison Test is 冘 b n , where
sn 3 n 3 兾2
3n 3 3n 3 3n 3 兾2
V EXAMPLE 3 2
n苷 兺 ne 1
Since the integral 2 x
1 xe dx is easily evaluated, we use the Integral Test. The Ratio Test
also works.
EXAMPLE 4
n苷 兺 1 n 4 1
Since the series is alternating, we use the Alternating Series Test.
EXAMPLE 5
k苷 兺 1 k !
Since the series involves , we use the Ratio Test. k !
EXAMPLE 6 兺
n苷
Since the series is closely related to the geometric series 冘 1 兾3 n , we use the Comparison
Test.
11.7 EXERCISES 1–38 Test the series for convergence or divergence.
n苷 兺 1 n
n苷 兺 兺 1 n 3 2n 2 5
1. 兺 n
n苷 1 3 n苷 1 n
2n
3. 兺 n n
4. 兺 n 2 23. 兺 tan 共1兾n兲
2 n苷 1 n苷 兺 1
24. n sin 共1兾n兲
n苷 1 2 n苷 1 n
n 5. 2 兺 n 6. 兺 25. 26. 1 n苷 1 n苷 1 兺 n 2 n苷 1 e n苷 兺 1 5 n
n苷 兺
兺 2 n k苷 1 k
k苷 兺 1 3 28. n苷 兺 1 n 2
9. 兺 2 k e 10. n 2 e 3
k苷 1 n苷 兺 1 n
29. 30. j sj
n苷 兺 1 cosh n
j苷 兺 1 5
n苷 兺 2 12. n ln n n苷 兺 sin n 1 5 k
k苷 兺 1 3 k 4 k
共n!兲 n
n苷 兺 1 n 4n
sin 2n
n苷 兺 14. 1 n ! n苷 兺 1 n
1 15. n !
sin 共1兾n兲
n 2 33. 34.
n苷 兺 0 n苷 兺 1 n 3 n苷 1 sn
n苷 1 cos n
17. n 2 1 兺 兾n
18. 兺 35. n苷 兺 36.
n苷 兺 2 共ln n兲 ln n
n苷 1 n苷 2 sn
19. n 兺 ln n 20. 兺 5 k
37. ( n s 2 1 n )
k苷 1 5 n苷 兺 1 n苷 兺 1
38. ( n s 2 1 )
n苷 1 sn
SECTION 11.8 POWER SERIES