Advanced Calculus fi..

442 Advanced Calculus, Fifth Editionto dejne these functions for complex z. From these series, one derives the Euleridentity (Problem 4 below):e i ~ = cosy + i sin y, (6.60)or the more general relation:.?I ?cThis last equation can also be used as a dejnition of eZ for complex z, the seriesexpression being then a consequence of this definition. This is shown in Chapter 8.PROBLEMS1. Evaluate the limits:ina) b) limn-+w (1 + i)n3 - 2in + 3in' - 12. Test for absolute convergence and for convergence: cl+i na) (!+)+(w)2+.-+(T) +... b) Czl ninm ni" 00 1C) Cn=l ,,2+1 d) Cn=l3. Prove that the series (6.57),(6.58), and (6.59) converge for all z.4. a) Prove the Euler identity (6.60) from the series definition of eZ, cos z, sin z.b) Prove (6.61) from the series definitions of eZ, cosz, sinz.5. Use the series expressions (6.57),(6.58), (6.59) to prove the identitieserz +e-lzerz - e-'za) cosz = 7 b) sin^=.^C) e Z ~ +22 = ezt . eZZ d) sin (-Z) = - sin ze) cos (-Z) = cos z f) sin2 z + cos2 z = 1g) cos 22 = cos2 z - sin2 z h) sin 22 = 2 sin z cos z6. Show that the following series converge for lzl < 1 and diverge for lzl > 1:2a) z+%+...+$+... b) 1+z+z2+...+zn+..-*6.20 SEQUENCES AND SERIES OF FUNCTIONSOF SEVERAL VARIABLESThe notions of sequence and series of functions extend at once to functions of severalvariables. Thusis a series of functions of the two variables x and y. The notion of uniform convergencealso extends at once, as well as the M-test and the properties described inSection 6.14.t : i