D. J . NEWMA N
' . *
THEOREM
1.1 . With P
nk
= (l/2*" ! )r
/ ( n +
^
) / 2
(^), we have Pn
k
/4eEn(xk)
THEOREM
1.2. R
n(xk)
2/n (in fact R
x n
{xk)
2/n).
We feel oblige d at this point t o explain th e connection betwee n th e precis e statemen t
of Theorem 1.1an d the crude earlier statement referrin g to th e requiremen t tha t n be much
larger tha n \fk, Thi s connection i s established onc e w e recognize Pk
n
a s a probability,
namely th e probability that , when a fair coin is tossed k times , the difference betwee n th e
number of heads and th e number of tail s should exceed n
7
in magnitude. Fo r large values
of th e parameters, then, the various limit theorem s of probability theor y apply t o our Pk
n
2
and give th e approximation Pk
n
^ (2/V?)/° ° r-r et ^ dt. Thus , as we promised, good
proximity i s obtained if and only i f n/y/k i s large.
We begin by giving the proof o f Theorem 1.2. No t onl y becaus e i t is the shorter one,
but, what i s more i n line with our current spiri t of humor, it turn s out simpl y t o be an ex-
ercise in elementary calculus . (O f course ther e i s calculus and calculus! Thu s the problem
of maximizin g 2~ x + 2~ {fx ove r (0, °°) is merely a n exercise i n differential calculus , but
beware, it doe s require a good bi t of finesse.)
PROOF O F THEOREM
1.2.W e find it easier to work with th e interval [0 , 1] rathe r
than [-1, 1], but this switchover is easily accomplishe d b y th e substitution o f
x2
fo r x.
To prove, then, that RQ
n
(xk)
2/n w e are required to produce a n S(x) €. Un fo r
which | | l/S(x) - x
k\\
2/n. Ou r choice is , naturally enough , the partial su m of th e power
series for
x~~ky
about th e point 1.
Hence, by the formul a fo r th e remainde r term , we have, identically,
Clearly S(x) :X~~ k an d so we are simply require d t o prove tha t
1 2
0-4) SOcj ~
xk
n throughou t [0 , I}) ,
but we shall prove more, that in fact
(1.5) ^k-^^-iwT
1.[0,11no)
S(x) n \2k + n J
Now introduce th e notation
(1.6) «-(f)* . Z = **, e = l(^y\ /(Z)=J-*'/*)«* ,
So that our formula, (1.3), simplifies t o
1 - CHZ\
(1.7) S(x) = ^r^ A C a constant.
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