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By I. J. Schwatt
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Extra resources for An Introduction to the Operations with Series
Whether ti be even or odd. 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 .
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 /n n + is/3) . and (22). )*© = g( (vii) The Applying i )i ( 2 ( 2* r0 - 1)^P + = - x) ^ ( [i " 1 ) ] 2 '" n -<- < . /2»-l , ( Applying n-k = k\ <5 (49).
Of (83) to find (10), which is the expansion of By means Letting in ( 1 n. (1). yrrV' ! a^- n a2n ~ k Now from follows fr 0m (^) (m . * =A and from + " ^ *) £ = * » i hence ^ = *• (125) . Jt then ^) = (K ^),that^[^]. More J^Lo^^^Qt-i^-^-"^-- (126) OPERATIONS WITH SERIES 18 n-k = k\ Letting y= then which the same as Given is (ii) where To m and ^? are find ±x» p is g (% * w" J( m any real negative, we have i we may write a positive integer, then from is is „ , + l *r > (130) ma W> may assume. L ( ) it follows a^— ~ TO , and that Inen S-^^'-'KOS^-^T)- ,,3,) y=a^(I + z TO )» (132) m and where By i „ value a the smallest Given (iii) (128) numbers.