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

By I. J. Schwatt

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83) u — cos x, we have (39) -g (-irsec^+^gcos^gQ, by Ch. I. (40) (97). A:=a = ((^a+1 ))[(l+«) n+1 -( 1 + a a; ) ] (41) Va+l * Stern, Journal fur Mathematik, vol. 79, pp. r)=secha: in terms of A= 6 67-98, finds by actual differentiation the higher 2Z 2 fex -e-*\ 2 to +C , Z= [ \6 -r C ) and / —— B=— e e - 5040Z ), (') (a;) = 5(1385 - 7266Z + 10920Z FW(z) =A(1385 -24568Z + 83664Z 2 - 100800Z 3 +40320Z 4 ), J 1 2 3 but does not give a general form of the higher derivative. Shovelton, Quarterly Journal of Mathematics, vol.

53) 1 since tan^fljje^ = 1, for /3 =0 only. And sec . =0. a; Jz=0 To reduce (53), we shall show that 2a *=2(f)(*i=6 \ u/ For, if k is 2n y S a =2 : ) (a)(^ 2a) 2n - ( 54 > even, 5 -g(J)(*- 2a)i * + ? Letting in the second summation k - a = 5=2 ( a', *" (a)(*- 2a > 2n then 2a)2 "' a§(«) If & is odd, the upper limit of Then, by means of (53) and S A;- 1 is (54), —^- we find 1 1 -- s-'^g^ff:. )! (>-*>••• This result can also be obtained by letting a; =0 in (44). m OPERATIONS WITJI SERIES 34 (ii) Another form of the nih derivative and the expansion of sec x is arrived at in the following manner.

Letting in (83) u — xnl If GO |? are any real numbers. Leibnitz's theorem, Applying (129) and to (133), dn we obtain y _n\y n°w Si= 2 ( " i)s 0(T) =0, if *

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