270 CHAPTER 5 Series and Differential Equations
asymptotically the series
∑∞
n=0
bnis no larger thanRtimes the con-
vergent series
∑∞
n=0
an.)
(ii) Let
∑∞
n=0
anbe adivergentseries of positve terms and let
∑∞
n=0
bnbe
a second series of positive terms. If for someR, 0 < R≤∞
nlim→∞
bn
an
=R,
then
∑∞
n=0
bn also diverges. (This is reasonable as it says that
asymptotically the series
∑∞
n=0
bn is at leastRtimes the divergent
series
∑∞
n=0
an.)
Let’s look at a few examples! Before going into these examples, note
that we may use the facts that
∑∞
n=1
1
n
diverges and
∑∞
n=1
1
n^2
converges.
Example 1.The series
∑∞
n=2
1
2 n^2 −n+ 2
converges. We test this against
the convergent series
∑∞
n=2
1
n^2
. Indeed,
nlim→∞
( 1
2 n^2 −n+ 2
)
( 1
n^2
) =^1
2
,
(after some work), proving convergence.
Example 2. The series
∑∞
n=0
√^1
n+ 1
diverges, as