5.3 SUMS AND PRODUCTS 161valid if and only if Irl < I. This is a simple consequence of the formula for a finite
geometric series.
There are many ways to determine whether a given series converges or diverges.
The simplest principles, however, are
• If }:ak < 00 (Le., the series converges) and the ak dominate all but a finite
number of the hk (Le., ak 2 hk for all but a finite number of values of k), then
}:hk < 00 •
• Likewise, if }:ak = 00 (Le., the series diverges) and the hk dominate all but a
finite number of the ab then}: hk = 00.
In other words, the simplest strategy when dealing with an unknown infinite series
is to find a known series to compare it to. One fundamental series that you should
know well is the harmonic series
1 I I
1+ 2 + 3 +
4
+ ...
.
Example 5.3.4 Show that the harmonic series diverges.
Solution: We will find some crude approximations for partial sums of this series.
Notice that
1 I 1 I^2 I3 + 424 + 4 = 4
=
(^2) '
since 1 and! both dominate !. Likewise,
and
In general, for each n > 1,
1 1 1 1 4 1
5 +"6+^7 +^828 =^2
1 1 1 8 1
... +
>
9 10 16 - - 16 =-^2.1 I 1 1
2 n + 1
+
2 n + 2
+ ... +
2 n + 2 n(^2 2)
'
since each of the^2 n terms are greater than or equal to 2n�I' Therefore, the entire
harmonic series is greater than or equal to
which clearly diverges.
1 1 1^1
1 + 2 + 2 + 2 + 2 +""
The key idea used above combines the obvious fact that
1 1
-<
a-h
with the nice trick of replacing a "complicated" denominator with a "simpler" one.
This is an example of the many-faceted massage tool-the technique of fiddling with