Advanced High-School Mathematics

272 CHAPTER 5 Series and Differential Equations

nlim→∞

1

(ln(n)^2

1

n

) = nlim→∞ n

(lnn)^2

= xlim→∞

x

(lnx)^2

l’Hˆopital= lim

x→∞

d

dxx

d

dx(lnx)

2

= xlim→∞

x

(2 lnx)

l’Hˆopital= lim

x→∞

d

dxx

d

dx(2 lnx)

= xlim→∞

x

2

=∞.

This says that, asymptotically, the series

∑∞

n=2

1

(lnn)^2

is infinitely larger

than the divergent harmonic series

∑∞

n=2

1

n

implying divergence.

The next test will provide us with a rich assortment of series to test
with. (So far, we’ve only been testing against theconvergent series
∑∞

n=1

1

n^2

and thedivergentseries

∑∞

n=1

1

n

.)

Thep-Series Test. Letpbe a real number. Then

∑∞

n=1

1

np







convergesif p > 1

divergesif p≤ 1.

The p-series test is sometimes called the p-test for short; the proof

of the above is simple; just as we proved that

∑∞

n=1

1

n^2

converged by

comparing it with

∫∞

1

dx

x^2

(which converges) and that

∑∞

n=1

1

n

diverged by

comparing with

∫∞

1

dx

x^2

(which diverges), we see that

∑∞

n=0

1

np

will have