i) Explain why the Taylor series for cos z and sinz (about z = 0) as...

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i) Explain why the Taylor series for cos z and sinz (about z = 0) as given in the text are not in standard form. Use these series to derive the standard form series for cos 2z and sin 2z. Note that one way to express the alternation in sign is to use the notations (1) n/ for n even, and (-1) (n+1)/2 for n odd. Recall the trig identity cos 2z = 1 - 2 sin z. Obtain the Taylor coefficients up to n = 4 of 1 - 2 sin z to give a verification for your cos-series from above. ii) The Fibonacci sequence {an} can be defined by the recursion relation an = an1 + an2 with a = 1 and a = 1. You are given that these Fibonacci numbers define an analytic function s(z) by a Taylor series 8(2) = an. 2 . n=0 and has a non-zero radius of validity. Explain why the functions zs(z) and zs(z) will also be analytic in this circle, and use index shifts to express them both in standard Taylor form (be careful with the summation limits). The recursion formula then suggests an arithmetic of series that will let you determine an explicit formula for s(z). Confirm your work with W-a. Finally, what is the largest radius of validity for s(z)? A google search on the decimal value will reveal its classical origins. i) Explain why the Taylor series for cos z and sinz (about z = 0) as given in the text are not in standard form. Use these series to derive the standard form series for cos 2z and sin 2z. Note that one way to express the alternation in sign is to use the notations (1) n/ for n even, and (-1) (n+1)/2 for n odd. Recall the trig identity cos 2z = 1 - 2 sin z. Obtain the Taylor coefficients up to n = 4 of 1 - 2 sin z to give a verification for your cos-series from above. ii) The Fibonacci sequence {an} can be defined by the recursion relation an = an1 + an2 with a = 1 and a = 1. You are given that these Fibonacci numbers define an analytic function s(z) by a Taylor series 8(2) = an. 2 . n=0 and has a non-zero radius of validity. Explain why the functions zs(z) and zs(z) will also be analytic in this circle, and use index shifts to express them both in standard Taylor form (be careful with the summation limits). The recursion formula then suggests an arithmetic of series that will let you determine an explicit formula for s(z). Confirm your work with W-a. Finally, what is the largest radius of validity for s(z)? A google search on the decimal value will reveal its classical origins.