Question

9,17,33 and ill like

cos 22 14. 2+1 In Exercises 1-26 find the Taylor series for the function about the given point. In each case determine values of z for which the series converges to the function. 3 1. + about zo = 0 2. 1+2 about = 1 3. (1 - 2)2 about 20 = 0 4. e* about 20 = 1+i 5. sin z about 20 = i 6. cos z about zo = 2 - 7. sinh z about zo = 0 8. cosh z about 20 = 0 9, 2: about zo = 1 10. sin?: about zo = 0 11. (2 + 1) cosh 3 about zo = 0 12. sin about zo = 7/4 13. cosh z about zo = i about 20 = 0 15. Cos (22-1) about 20 = 0 16. sin (322 +2) about 20 = 0 17. 112. about 20 = 3 about 20 = isoler 19. ( _ 2-2 DO about zo = 0 40. 1+2 about 2 = 1 + 21. sinh 42 about 0 = =ị 22. Log z about zo = 23. Tan-12 about zo = 0 24. 3-d about 20 = 2 25. V1 +23 about 20 = 0 26. Sin about 20 = 0 In Exercises 27-32 identify all zeros and their orders for the function. 22+1 28. 23e3+2 22 24. 27. 2(2+1) 29. sin 32 30, sinh? % +51 31. 32. Log (2-1) 33. Use series to prove Euler's identity 1.29. e-cosesinei

Answer 1

Q.9, set f(2)= 22 ZEC. Then fee) 2822, e 22-2+2 2 (2-1) [e1125 e [1+ 212-1) + (2(2-1) 5+ {212-15 se 22- 22 ] ( Nu u V 2 e 2 no ! which is convergent for every Z, as 22 is an entire Fernetion.

+27 Q.17 we have f(2)= st. 75-1/2 The Taylor sersies expansion of f about 20=3 is given by f(3) = 3 from ) (2-3)? --- 0). Now f(3) = ha f'(2) -- izzi f() =2233 fin (2) = (-113 3.84 m in general fm (2) = (-13 M. 27. (1+22) MTI wo. I'm) (3) = (-1). Min Flot - (-)" (3)"? so, from (1) we have #(2) *(-1)" ()"(2-3). n. 2n 20 This series is convergent for 13 (7-2) | <1 => 12-31 < 712. So, 121= 12-3+3) = 12-31+3<2,3 23 je 121 < 13/2. 27

2-33. Let f(0) = ei. f210) = coso f3 (O) = sino Each of the functions fi fq, ts are different over R (the set of real numbers). Therefore each of the function Ras a Taylon sessies representation. we considers the Taylor sersies about Oo = 0. so. Ez (0) = { fontoo) on, for any OER nai But by 2n) (O) = (-1)" coso e torne) = 1) ind so. P2?^)(0) =(-19 e f 2M-) (0) = 0. From (1) fa (O) = ¿ 12 0 029) - oen nas (an)! = 1- en + en - 2007 Again, Taylors Persies of tz about do=0. 13(0)= La to 100) on=& fonico) en 13 non nro 0 (21) n. (2n-D co F = E (0)= (-) in 22n-1 f 54 (21-110 nap! facere) ---MESO L E COM 2n-15, 02n1 net 05-Otance =0-357 Now, f(0) = Rio 10) fmoon I n (o)= ime ir - )0) in non = 1 +8 -10 +04 1993 -(-+- +i10 = coss risino. . Therefore ero coso eising