An Introduction to the Operations with Series by I. J. Schwatt

By I. J. Schwatt

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Applying (33) to (29) and We shall express cos Now taking the ^ and ^ sin sum of (27) C03 and (35) to (30) gives the values of and (35) 8 and Sv also as summations. (28), T=2(72r[( we have 1+i''" + ( 1 -^ < 36 > their difference gives [2=1] p] C+0-=g(-D*(a + i But and (1 -i)»=g(- then by means of (38) and g (-D'G^O ^ffl-^jV (39), we tt7T l) obtain from (36) and (37) 1 COS and sin Applying mr cos-^ and (40) to (29) way we In a similar . nir + 1 sin -j = 1 2 1 (41) to (30) gives (21) obtain by means of .

1)*( 2 ^ )- (53) SERIES OF BINOMIAL COEFFICIENTS 2. (i) To 47 find the value of S= Now =((**-*)) £ a +*)*-* k=0 (ii) To We have find the value of flf-g ( l)*" 1 ^ _j) -(- 1)" (58) n+l =S ((^))(i + *)1 (iii) To sum ((* B+1 --*))(i +»)» - ( - i) n l) n = ((*»)) -f; 1 )- ("I)"- (59) the series frHs^K^Wii) Applying (61) and (62) to (60), = (_l)j>-«-m ^ (62) we have ((x*~ B))(l +a;)-»- 1 ((a5«'-*-«))(l +a;)-«- 1 (64) jfc=n = t _i v- f+ \ = r n-„(-»-»-2) p-n-m J (\p-n-mj 1 ; \ x/ (65) ) OPERATIONS WITH SERIES 48 (iv) To find the value of ^= Since *s #=0, a polynomial in n of degree m, if m

Y-^i=w+i + ^i»\L. l2n-l x expanding y = x cosec x * is /t the following 132 ) DERIVATIVES OF TRIGONOMETRIC FUNCTIONS and by means 41 of (85) ^i Mi:)^^ m^M^ yn a % Applying Ch. I. (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.

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