Hello! Due to the chegg image system problem, i have uploaded the question @ htt

Question

Hello! Due to the chegg image system problem, i have uploaded the question @ http://imageshack.us/scaled/landing/197/bax8.png

First of all, z is a variable consisting of complex numbers.

Question is:

Let F(z) be the anti-derivative of the function f(z) = cos z^3  with F(0)=0.

Express F(z) as a power series around z = 0, giving both the first three non-zero terms and general (n th) term.

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Let F(z) be the anti-derivative of the function f(z) = cosz3 with F(0)=0. Express F(z) as a power series around z = 0, giving both the first three non-zero terms and the general (n th) term.

Explanation / Answer

F(z) be the anti-derivative of the function f(z) = cos z^3

so F'(z) = cos z^3 = 1-(z^3)^2/2!+(z^3)^4/4!-(z^3)^6/6!+......

= 1- z^6/2! + z^12/4! - z^1/6! +....

F'(0) = 1

F"(0) =F"'(0) = F(iv)(0) = F(v)(0) =F(vi)(0) = 0

F(vii)(0) = -6!/2! = -720

F(xiii)(0) = 12!/3! = 7257600

so F(z) = F(0) + zF'(0) + z^2/2! F"(0) +......+z^7/7! F(vii)(0) +.....+ z^13/13!F(xiii)(0)+.....

F(z) = 0 + z - 6!/7!2! z^7 + 12!/13!3! z^13 = z - 1/7.2! z^7 + 1/13.3! z^13 (first 3 term)

general term

F(z) = (-1)^(n+1) z^(n+5(n-1)) / (n+5(n-1))! n!

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