Page 263 - Calculus Workbook For Dummies

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Chapter 13: Infinite Series: Welcome to the Outer Limits

Apples and Oranges . . . and Guavas:

Three Comparison Tests

With the three comparison tests, you compare the series in question to a benchmark
series. If the benchmark converges, so does the given series; if the benchmark
diverges, the given series does as well.

The direct comparison test: Given that 0 # a n # b n for all n, if ! b n converges, so
does ! a n , and if ! a n diverges, so does ! .
b n
This could be called the well, duhh test. All it says is that a series with terms
equal to or greater than the terms of a divergent series must also diverge, and
that a series with terms equal to or less than the terms of a convergent series
must also converge.
The limit comparison test: For two series ! a n and ! , if a n > 0, b n > 0 and
b n
lim a n = L, where L is finite and positive, then either both series converge or
n " 3 b n
both diverge.
The integral comparison test: If f xh is positive, continuous, and decreasing for
^
3
3
f nh, then !
f x dx either both converge or both
all x $ 1 and if a n = ^ a n and # ^ h
n 1 1
=
diverge. Note that for some strange reason, other books don’t refer to this as a
comparison test, despite the fact that the logic of the three tests in this section is
the same.
Use one or more of the three comparison tests to determine the convergence or diver-
gence of the series in the practice problems. Note that you can often solve these
problems in more than one way.

3
Q. Does ! 1 converge or diverge? A. It diverges.
n 2 lnn
=
Note that the nth term test is no help
1
because lim = 0. You know from the
n " 3 lnn 3 3
p-series rule that ! 1 diverges. ! , of
1
n
n
=
n 1 n 2
=
course, also diverges. The direct compar-
3
ison test now tells you that ! 1 must
=
n 2 lnn
diverge as well because each term of
3
! 1 is greater than the corresponding
=
n 2 lnn 3
1
term of ! .
n
=
n 2