538 Chapter 8 • Infinite Series of Real Numbers
(d) Vn EN, E(n) =en, and Vm E Z , E(m) =em.(e) Vn EN, E(l/n) = e^1 fn, and Vr E Q, E(r) =er.
(f) x --+oo lim E(x) = +oo; x--+-oo lim E(x) = O;
(g) The range of E is (0, +oo).Theorem 8.8.7 The function E(x) is identical with the function ex defined in
Sections 5.6 and 7. 7. That is, Vx E JR, E(x) =ex.
Proof. Apply Exercise 5.1.29. •Theorem 8.8.8 The function ex has an inverse.
Definition 8.8.9 The inverse of the function ex is (temporarily) called L(x).
Theorem 8.8.10 The function L(x) is identical with the function lnx defined
in Sections 5. 6 and 7. 7.
TRIGONOMETRIC FUNCTIONSDefinition 8.8.11 We define the functions S : JR, JR and C : JR, JR by
oo (-l)kx2k+1^00 (-l)kx2k
S(x) = ~ ( 2 k + l)! and C(x) = ~ ( 2 k)!.(These series converge absolutely Vx E JR.)Theorem 8.8.12 The functions S : JR, JR and C : JR, JR have the following
properties:
(a) Vx E JR, S(-x) = -S(x) and C(-x) = C(x).
(b) S(O) = 0 and C(O) = 1.
(c) S(x) and C(x) are differentiable everywhere, and Vx E JR,
{1} S'(x) = C(x) and C'(x) = -S(x);
{2} S"(x) = -S(x) and C"(x) = -C(x).Theorem 8.8.13 (a) Vx E JR, S^2 (x) + C^2 (x) = 1.
(b) Vx E JR, IS(x)I '.S 1 and IC(x)I '.S 1.(c) Vx E JR, S(x + y) = S(x)C(y) + C(x)S(y);