-39~

x
Expand ['(x) = %e YJ“{II[I+-;]¢ N

» Using formula

(17') for the reciprocal, (page 27), we obtain

e 2 8.8
P =3 -ye+ Lo Tt e 14 o = Foya® 4 G0y 4§ 0400 +or ]

i P1o)E - 20,)%
=;'Y+(Y 022 ey f3Y°z+ 3/37

4
+ (y* + 6y20, + 8y0y 4 60, + 369')%. I

This result agrees with the rule stated below Theorem IX which,
applied to the D function, says that by replacing each Y apd o by
= Y and - 0 respectively, in cne sekies, we obtain the other,2®

This is equivalent to making all the signs positive inside the

parentheses in formula (12) and wultiplying Cy by (=1)¥ ., Using
the decimal approximations of ¥ and O, we get the approximations
P(x) = 201 = 5772x +°.9361x" - 1.81154° + .8928x% «eoes g

Ffl—iz #11+ .5773s - .6569x" - ,0403x® + .2EGEx4 esvss ]
x

- X
Expand the product sinx = x I '[1-;;-] M This is

X
absolutely convergent if " g kept with its factor. Then the

factors can be paired sc that sinx= x1II [1 - (;,5,7)’] . In this
1

29
form, s; = (—1)’—(—2—"—)_——5_4 » where By is the jth Bernoulli number.?®
2(25)!

2
Replacing each x in the formulas by -7’—:-; and using fermulas (11) and

(12), we obtain

3 R

2 n? 1 ap % n*
C;:wflfljz—fi, Co = =[m*B, —-T]zg’ Cs="7", weese
€

2
J 1 x x % e
Now s1nx=x[1+0,;—+ca;—+ca;‘7+ v ]

M«————————-——————-—w

e These resulte agree with those 2iven by Nieleen, sec. 15.
2

" Price, p. 145. Crelle, Vol, 20, re 11, Vol. 85, p. 269, for tablee of Bts,