I need help on an assignment on power series, taylor series, and fourier series!

Appreciate it if you may show me how to.

1 Power Series - practice Determine the largest open interval on which each of the following power series converges: (a) 23:0 \"356" (b) Eveline\"23" 2 r ('3) 22:0 2:. 3\" (d) 221002? - U3" 2 Taylor Polynomials - practice Approximate the value of each of the following functions at r = 1 using its cubic Taylor polynomial T3 centered at r = 0: (a) ziti (b) rel (c) 24 + 213 - 912 +4 3 (d) cos(I) (e) cos(10r) Would you expect your answer for part (d) or (e) to be more accurate? Write a sentence explaining why:3 Fourier series - practice Let f(t) = sin't for each t E R. f has a period of 27, so we can calculate its Fourier series. When calculating Fourier coefficients, to can be any real number, so you should choose whichever one you think will be most convenient. (a) Calculate ao = 2. fo f(t)dt. The trig identity sin? t = 1-cos(21) 2 may be useful. (b) Calculate a1 = 4 flot2x f(t) cos(t) dt. (c) Calculate by = = Sto+2x f(t) sin(t)dt. The trig identity sint = 1 - cost may be useful. (d) Calculate a2 = 4 flotzx f(t) cos(2t)dt. The trig identity mentioned in (a) and the trig identity cos? u = 1+cos(2w) may be useful. (e) Explain why all of the Fourier coefficients other than the four you calculated above will be zero. (Hint: Assume all the other Fourier coefficients are zero, write out the Fourier series do + En= (an cos(nt) + on sin(nt)) explicitly, and explain why it equals f(t).)