Advanced High-School Mathematics

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