7.6 The Fundamental Theorem of Calculus 4151
(b) \:/x E (0, 1), 0 < 'l/Jn(x) < 1.
n.( c) 'I/Jn ( 0) = 0 and the successive derivatives of 'I/Jn ( x) at 0 are:'lj;~m)(O) = { :,! c., ~~! : : :~,}
0 ifm>2n.
In fact, 'lj;~m)(x) = 0 for all x, ifm > 2n.
Proof. Exercise 19. •Theorem 7.6.21 ex is irrational, for all nonzero rational numbers x.
*Proof. Suppose^12 x is a nonzero rational number. If e-x is rational, so
is ex. Thus, it is sufficient to prove the theorem for positive rational numbers
x = !!. , where p, q E N. Further, if ex is rational, so is eP = (ex )q. Thus, it is
q
sufficient to prove that eP is irrational for all p E N.
For contradiction, suppose 3 p E N 3 eP is rational. Then 3 a, b E N 3 eP =
~-For each n EN, let 'l/Jn(x) be defined as in 7.6.19, and define Fn(x) by
Fn(x) = p2n'l/Jn(x) _ p2n'lj;~(x) + P2n-2'1j;~(x) _ ... _ p'lj;(2n-l)(x) + 'lj;(2n)(x)
2n
= ~)-l)kp2n-k'lj;~k)(x).
k=O
One can easily show (Exercise 19) thatd~Fn(x) = -pFn(x)+p^2 n+l'l/Jn(x).
Thus, Fn satisfies the differential equation
F~ + pFn = p2n+l'l/Jn·
If we multiply both sides by ePx, we have
ePx F~ (x) + pePx Fn(x) = ePxp2n+l'l/Jn(x)
i.e., d~ [epx Fn(x)] = ePxp2n+1'1/Jn(x).- See Theorem 8.8.7 for an easier proof using infinite series.
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