By I. J. Schwatt
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Additional info for An Introduction to the Operations with Series
Or From (3) and (4), we then (4) obtain ( (ii) Let To 5) find the value of ^-(-^(-D*^); 42 (8) SERIES OF BINOMIAL COEFFICIENTS then «+«k-g<-iy(jf)-(»-'l)»-0 and 8-S1 =(-iy(^} The results Letting in (9) (10) ^O^k^i) Therefore (1 1) and (12) might n-Jc = k', (7), also be obtained as follows <"> : then ^(-IJ-gC-l^J =(- n i) 43 (13) i B 2n i;((^))(i +*)- ((« -*))(i +*) A:=0 = (-l)»((*«))(l+a;) 2*-i 2n-l\2% y v which is the same as (11). (HQ To show that (-l) n /27i 2 \7i J2n In a similar way (12) is obtained.
67) to (137), we (i37) obtain x cosec 2 t(-l) a G -" a )«2n - d38) CHAPTER III. SERIES OF BINOMIAL COEFFICIENTS. In the preceding chapters we have had occasion to reduce Binomial Coefficients find the value of a series of them. We shall give here a few examples will illustrate additional methods of the operations with Binomial and to which Coefficients. 1. (i) To find the value of -Jift> 2n 2 2 ^ + ^ = f;( ^) = (l + l) 2w = 2 2". then Letting n - k = h' in (3) we have (2), 2 ^|ff)=s©-( ;) S-S^Cf).
1- (109) powers of cosec x and cot x. Let w = sin$; then, by Ch. 1. tj(*) ( |; [J"" where ' ( parts, (113) ^ - ( 113 > (114) becomes (us) _ o~ v (" 1 >\%0 ) (- 1 P ) cot20a; &+l) cot2ma; ( - 116 > (117) . DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 39 Therefore d 2n 2» -\) k /2n + l\ /k\ ^cosec = (-l)"gl^(^ /)2Q(i-2a^cosec»-«« ( k a: i>f2 ,, (118) ^(^(^-Sa^+icosec^+^i^/s+i. ] (119), we obtain ( l -^(i:i)| 0 o^^— s <-^; 0=0 where (ii) y Another form yx = I. >» where and, by Ch. ^ = l-(-l)' for the higher derivative of y Now cot2fl+ 7) e 2ix _ : < 121 > (62) and i (83), Hence Now, applying to (124) the method by which we have from (121), (64) was obtained from (63), fin,, n pZiax a /_\ ^=^^(^^^(-1)^(3(1+2^.
An Introduction to the Operations with Series by I. J. Schwatt