460 Chapter 8 • Infinite Series of Real Numbers
00
function such that lim f(x) = 0. Then the series L an converges ¢==:::} the
X--+ (^00) n=no
improper integral Jno roo f converges.
00
Proof. Suppose L an and f are as described in the hypotheses. Since f is
n=no
monotone on [no, + oo), f is integrable on any compact subinterval of [no, +oo).
Let n be any integer > n 0 , and consider the partition
P ={no, no+ 1 ,no + 2, · · · ,n}
of [n 0 ,n]. Since f is decreasing on [n 0 , + oo), we see from Figure 8.l(a) that
n-1 n - 1
I: ak = I: J(k) = S(J, P) ~ J;: 0 f.
k=no k=no
y
n- 2 n-l n x
(a)
y
n-2 n-l n x
(b)
Figure 8.1