Calculus: Analytic Geometry and Calculus, with Vectors

(^648) Series
and k are quite small, it is advisable to substitute in (3) and simplify the result
before making numerical calculations. Thus we can put (3) in the form
1 k)
(5) log 2n (k/ = log 2 - `k +
1

1. log
   2k
   n
   -(n-k+1)log2(1 )+En.k,
   where En,k = En - Ek - En_k Much progress in probability and statistics is
   based upon the idea that when k is near n/2, we can represent it in the form
   (6) k=2+xv,
   where X is a number that depends upon k and n. When (6) holds, we find that

(7) 2k =1+2(1-n)=1-

and we can put (5) in the form

(8) log 2n (k) =

log

nar

and hence

(9)

-(?+a-v/,-

+1)log(1+

-(2-A V1- +Y1) log 1 2X

log Zn (k) =log

n

2 1 log (1 4n/

-Ay7GLlog(1+2X-)-log(1 2 +E_,k.

Vn

Now let a be a positive number and suppose that k is close enough to n/2 to

make

(10) 2-a'<k2 +X /n52+aVn.

Then Jai < a and we will have

4X2<

n

2X = <1

n

provided n is sufficiently great, say n > na. It can be shown that there is a

constant MI such that, when jx( < I,

(12)

(13) log (1 + x) - log (1 - x) = 2x + F2(x)lxIa,

log (1 - x) = -x + FI(x)Ix12

where JFI(x)l 9 MI and JF2(x)l MI. These facts and (9) yield the conclusion