8.3 Series with Positive and Negative Terms 479
and
0 < S2n+1 - S = azn+2 - azn+3 + azn+4 - azn+5 + azn+6 - · · ·
< azn+2· (11)
Putting (8) and (10) together, we have S2n < S < S2n+ 1 , and putting (9)
and (11) together, we have IS - Sml < am+I regardless of whether m is odd
or even. •
oo (-l)n+l
Example 8.3.5 Prove that L 2 converges, and find its sum to
n=l n + 5n+^2
three decimal place accuracy.
Solution. The sequence { n 2 +~n+z} is a monotone decreasing sequence
of positive numbers converging to 0 (show). So, by Theorem 8.3.2, we may let
- oo (-l)n+l
S = ""' 2
5 2
. By Theorem 8.3.4, we know that \:/n E N,
n=l Ln + n+
IS s I
1 1
- n < (n+1)2+5(n+1)+2 n^2 +7n+8·
To find S to three decimal place accuracy it is sufficient to calculate Sn for
n large enough that IS - Snl < 0.0005. For this it will be sufficient to take any
n such that
1 5 1
---.,.-----< ---= --
n2 + 7n + 8 10, 000 2000
i.e., n^2 + 7n + 8 > 2000
n(n + 7) > 1992.
By direct calculation we find that this inequality is satisfied when n :'.'.: 42.
Thus, S 42 will have the desired accuracy. Resorting to a calculator, we obtain
(to nine decimal places):
42 (-l)n+l
S42 = L 2 5 2 = 0.085371590.
n=l n + n +
Thus, to three decimal place accuracy,
s = 0.085.
In fact, according to Theorem 8.3.4, S 42 < S < S 4 3. Using a calculator, we
find S 43 = 0.085855618 (to nine decimal places). Rounding off, Theorem 8.3.4
(a) assures us that
0.0853 < S < 0.0859. D
