Question

3. The outcome of this process, illustrated for f(r) = cos(z), is to produce polynomials T "(r) in powers of r-a and a Taylor series Σ..a (z α)" where we have developed a precise fola for the a's in terms of the appropriate derivatives of f(z) evaluated at α. Write out that generic formaula for the α, . based on your work above. (Note: when α-+0, we often just use the simpler notation Pn(r) instead of T:nalr). and call the result the "Maclaurin poynomials/Maclaurin series.) Write down the formulas for the Taylor series at 0 for each of the following. Memorize them. (a) Taylor series of cris: (b) Taylor series of sin(r) is: (c) Taylor series of cos(z) İK (d) Taylor series of İlİİs: (e) Taylor series of In(1+z) is Calcuhis Apri 1, 2019 Taylor Series WA Name 5. The convergence of a Taylor series Taylor series Σ. 4(1-a)" (for any particular function f(z)) can be established by applying the Ratio Test to the series Σ﹂obk Σχο aa(z-a)" using 4 aa(z-a)" and computing ρ lim,mx 1 uil, when ρ 1 the test fails, and we must try other convergence tests for series. (a) Apply the Ratio Test to the Taylor series for cos(z) at = 0, and put down your outcome here. Do we need other tests? If so, apply those, too.

Answer 1

"F(x) = Cos x T_ma (x) = Sigma 6infinity_k = 0 a_k (x - a)^k write out generic formula for a_k S taylor series is a_k = f^k(A)/k! Taylor series at x = 0 (a) e6x = 1 + x + x^2/2! + x^3/3!.... (b) Sin x = x - x^3/3! + x^5/5!.... (c) Cos x = 1 - x^2/2! + x^4/4!.... (d) 1/1 -x = (1 - x)^-1 = 1 - x + 2x^2 - 3x^3... (e) ln(1 + x) = x - x^2/2! + x^3 - .... Ratio test for cos x at x = 0 Cos x = sigma^infinity_n = 0 (2n)! (-1)^n (x)^n \r\n f(x) = Cos x Cos x = Sigma^infinity_n = 0 (-1)^n/(2n)! x^n b_k = (-1)^k/(2k)! x^k Lim_n rightarrow infinity