Please show all your work, step by step, and show final answer. Answer must be c

Question

Please show all your work, step by step, and show final answer. Answer must be correct. Question 1,2 and 4. Thank you for your help.

Let f(x) = Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. If possible, determine the radius of convergence of the series. Choose a Taylor series and a center point a to approximate the number sin(0.997r) with an accuracy of at least 10-4. Evaluate the following limit using Taylor series. (You do not allow to use L'Hôpital's Rule to answer the question.) Let f(x) = for x plusminus 0 and /(0) = 1. Use the Taylor series for f and f' about 0 and evaluate f(2) and to find the value of

Explanation / Answer

1) The radius is the lim a(n)/a(n+1) = lim 1/(2n)!/ 1/(2n+2)! = lim (2n+2)(2n+1) = +infinity

2) you check with a calculator with n=1 you see that (0.99pi)^(2n+1)/(2n+1)! is bigger than 10^-4, you continue with n=2, etc, you'll see it works only from n>=7.

3) it is 1/20 , you are right i forget to multiply by 6 the 1/5! sorry

4) it is just the formula of d

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