Advanced High-School Mathematics

SECTION 5.2 Numerical Series 273

the same behavior as the improper integral

∫∞

1

dx

xp

. But, wherep 6 = 1,

we have

∫∞

1

dx

xp

=

x^1 −p

1 −p

∣∣

∣∣

∣

∞

1

=







1

p− 1 ifp >^1

∞ ifp < 1.

We already know that

∑∞

n=1

1

n

diverges, so we’re done!

The p-Test works very well in conjunction with theLimit Com-
parison Test. The following two examples might help.

Example 5.

∑∞

n=1

n^2 + 2n+ 3

n^7 /^2

converges. We compare it with the series

∑∞

n=1

1

n^3 /^2

, which, by thep-test converges:

nlim→∞

Ñ

n^2 + 2n+ 3

n^7 /^2

é

(

1

n^3 /^2

) = limn→∞n

(^2) + 2n+ 3
n^2

= 1,

proving convergence.

Example 6.

∑∞

n=1

n^2 + 2n+ 3

n^7 /^3

diverges. We compare it with the series

∑∞

n=1

1

√ (^3) n, which, by thep-test diverges:
nlim→∞
Ñ
n^2 + 2n+ 3
n^7 /^3
é
Ñ
1
√ (^3) n
é = limn→∞n
(^2) + 2n+ 3
n^2

= 1,

proving divergence.

There is one more very useful test, one which works particularly
well with expressions containing exponentials and/or factorials. This
method is based on the fact that if|r|<1, then the infinite geometric