12.2 Ratio test and integral test 603where 0 < C.(s) < 1. Letting n oc and using the definition
(12.272) (s) _
k=1kof the Riemann zeta function g(s) gives the nontrivial formula
(12.273) c(s) _ I1 k, =s 1 1 + C(s),
where 0 < C(s) < 1. The above results and the results obtained in the
problems at the end of this section imply that(12.281) I 1 < 00
k1ke(s > 1)(12.282)^1 = w
kmlP(s < 1)(12.283) +^1 < `c (s > 1)
k- 2 k(logk)°(12.284)^1 (s 5 1)
k2 k(log k)'.These series are often used with the comparison test to determine whether
other given series are convergent.Problems 12.29
1 Use the ratio test to show that the series in(a) es = 1 ++X2+38+44+
x2 x4 x8
(b) cosx=l-21 T-!-6!+
x3 xa x7
(c) Sj- y +converge for each x, and that the geometric series in(d) 1 1 x=1-}-x+x2+xa+...
converges when jxj < 1 and diverges when IxI > I.
2 Use the ratio test to show that the series1!x+2!x2+3x3+4!x4+ ..
diverges for each x for which x 0 0.