Math 133 Comparison Tests Stewart §11.4
Convergence and divergence. We continue to discuss convergence tests: ways
to tell if a given series P∞
n=1 an= limN→∞ PN
n=1 anconverges (to a finite value), or
diverges (to infinity or by oscillating).∗So far, we know convergence for two kinds of
standard series:
•Geometric series: P∞
n=1 crn−1converges to c
1−rif |r|<1, diverges if |r| ≥ 1.
•Standard p-series: P∞
n=1 1
npconverges if p > 1, and diverges if p≤1.
In this section, we test convergence of a complicated series Panby comparing it to a
simpler one (such as the above): a convergent ceiling Pcn, or a divergent floor Pdn.
Direct Comparison Test: Let Mbe a positive integer starting point.
•If 0 ≤an≤cnfor n≥M, and
∞
P
n=1
cnconverges, then
∞
P
n=1
anconverges.
•If an≥dn≥0 for n≥M, and
∞
P
n=1
dndiverges, then
∞
P
n=1
andiverges.
These results are clear, since the series P∞
n=1 anis term-by-term smaller or larger
than its comparison series, except possibly the first M−1 terms.†
Example: Determine convergence of:
∞
X
n=1
n−1
n2√n+ 1. We have:
an=n−1
n2√n+ 1 ≤cn=n
n2√n=1
n3/2for n≥1,
since on the left the numerator is smaller and the denominator is larger than on
the right. The comparison series P∞
n=1 cn=P∞
n=1 1
n3/2is a standard p-series which
converges, so P∞
n=1 analso converges.
Example: Determine the convergence of:
∞
X
n=1
23n+sin(n)
3n+ 4n2.
As a rough guess, we ignore the lower-order terms in numerator and denominator
to compare with 23n
3n=8
3n, which makes a divergent geometric series, so our
series anshould also diverge. However, it is not clear that anis really larger than
this comparison series, so we cannot use dn=8
3nas a divergent floor for anin the
second part of the Comparison Test.
We want to produce a fractional dnfrom our anby making the numerator smaller
and the denominator larger. To bound the numerator: 23n+sin(n)= 23n2sin(n)≥
Notes by Peter Magyar magyar@math.msu.edu
∗A general divergent series might oscillate up and down forever, but a positive series (with an≥0)
either levels off to a finite value, or diverges to infinity.
†Here we use the completeness axiom of real analysis, which states that if a series of partial sums
has an upper bound, sN=PN
n=1 an< B for all N, then the least upp er bound L= lim
N→∞
sNexists.
