Advanced High-School Mathematics

SECTION 5.2 Numerical Series 265

Series 2: 1 +

1

22

+

1

32

+

1

42

+···.

To see how the above two series differ, we shall consider the above
diagrams. The picture on the left shows that the area represented by
the sum 1 +^12 +^13 +··· isgreaterthan the area under the curve with
equationy= 1/xfrom 1 to∞. Since this area is

∫∞

1

dx

x

= lnx

∣∣

∣∣

∣

∞

1

=∞,

we see that the infinite series 1 +^12 +^13 +··· mustdiverge(to infin-
ity). This divergent series is often called theharmonic series. (This
terminology is justified by Exercise 20 on page 109.) Likewise, we see
that the series 1 + 212 + 312 +··· can be represented by an area that

is ≤ 1 +

∫∞

1

dx

x^2

= 1−

1

x

∣∣

∣∣

∣

∞

1

= 2,which shows that this series cannot

diverge to∞and soconvergesto some number.^7

5.2.1 Convergence/divergence of non-negative term series

Series 2in the above discussion illustrates an important principle of
the real numbers. Namely, ifa 0 , a 1 , a 2 , ...is asequenceof real num-
bers such that

(i)a 0 ≤a 1 ≤a 2 ≤..., and

(ii) there is an upper bound M for each element of the sequence,

i.e.,an≤M for eachn= 0, 1 , 2 , ...,

(^7) It turns out that this series converges toπ^2
6 ; this is not particularly easy to show.