674 Appendix C • Answers & Hints for Selected Exercises
(i) For x > 0, f'(x) = ~e-lfx. Take p 1 (x) = x^2. Thus, the statement is
true when n = l.
(ii) Suppose the statement is true when n = k. That is, J(k) (x) =Pk ( ~) e-^1 /x,
where Pk(x) is a polynomial with constant term 0. Then, Vx > 0, JCk+ll(x) =
Pk (~)·~e-llx+Pk (~)· ~J-e-1/x = ~e-1/x [Pk(~)+ Pk(~)] = e-1/xPk+l (~),
where Pk+ 1 (x) is a polynomial with constant term 0.
(b) We show that Vk EN, J(k)(O) = 0. First, note that lk)(O) = 0, since
f is constant on (-oo, OJ. We shall use mathematical induction to show that
Vk EN, Jlk)(O) = 0.
(i) Jl^1 )(0) = lim fCxt,_b(O) = lim e-~/x = 0 by Ex. 14 (a).
x-+xci x-+xci
(k+l) f(k) (x) f(k) (0)
(ii) Suppose true for n = k. Then f + (0) = lim~
x-+xci
= lim+ ~Pk UJ e-l/x. Now ~Pk ( ~) is a polynomial in ~ with constant term
X-+Xo0, so by Ex. 14 (c), lim ~Pk(~) e-l/x = 0 .. '. Jlk+l)(O) = 0.
x-+xt
( c) Since J(k) (0) = 0 for every k, the nth Taylor polynomial for f about 0
is Tn(x) = 0. But, Vx > 0, f(x) =f. 0. Thus, Vx > 0, Tn(x) f+ J(x).
- (a) 1 (b) -e/2 (c) -1/2
Chapter 7
EXERCISE SET 7 .1- By hypotheses, Va EA, a EB, so a~ supB. Thus, supB is an upper bound
for A.:. sup A~ supB. Similarly, Va EA, a EB, so a:'.". inf B. Thus, inf Bis
a lower bound for A. :. inf A :'.". inf B. - (a) Va E A , b E B , a + b E A+ B and a :'.". inf A and b :'.". inf B, so
a+ b :'.".inf A+ inf B. Thus, inf A+ inf Bis a lower bound for A+ B. Suppose
w is another lower bound for A+ B. Then Va E A, b E B, a+ b :'.". w; i.e.,
a :'.". w - b. Thus, Vb E B , w - b is a lower bound for A. Then,
Vb E B , w - b ~ inf A; i.e., w - inf A ~ b.
Thus, w - inf A is a lower bound for B, so w - inf A < inf B .. ·. w <
inf A+ inf B. Putting these results together, inf(A + B) =inf A+ inf B.
- (a) Suppose that 3K > 0 3 Ve> 0, x ~a+ Kc. Let€> 0. Then c/ K > 0,
so x ~a+ K (c/ K) =a+€. By the forcing principle (Thm. 1.5.9) x ~a.
(b) Suppose that 3K > 0 3 Ve> 0, x :'.".a-Kc. Let c > 0. Then c/K > 0,
so x :'.". a-K (c/ K) =a-€, so -x ~ -a+c. By the forcing principle, -x ~-a;
i.e., x :'.". a.