Calculus 2 Hour Exam 3 - Math 231 Spring, 2010

1.	Compute a power series representation for the function, f(x) = sin(x) / 2x. Express your answer using summation notation. Hint: Use the series expansion for sin(x). ∞ Answer: ∑ (-1)^n x^2n/ (2(2n + 1)! n=0
2.	a. Compute the first five terms (up to x^4) of the power series of -1/ √(1 – x^2 ) Hint: Use the binomial series expansion. b. Integrate your answer from (a) to obtain the first five terms (up to x^4) of the power series for cos^-1 (x) Answers: a. - 1 + ½ x^2 - 3/8 x^4 b. – x + 1/6 x^3 - 3/40 x^5 + C
3.	Find the radius of convergence of the series ∞ ∑ [(3n)! / (2n)!^3] x^n Hint: Use ratio test. n=1 Answer: R = ∞
4.	Compute lim (6 e^x - 6 - 6x - 3x^2 - 6x^3 ) / x^3 x → 0 Hint: Use the series expansion for e^x . Answer: - 5
5.	x a. Compute a power series representation for the function, f(x) = ∫ sin(t^2) dt Express your answer using summation notation. 0 b. Use the first 12 terms (up to and including x^11) of the power series from (a) to find an expression for 1 ∫ sin(t^2) dt and estimate the error. 0 ∞ Answers: a. ∑ (-1)^n x^4n + 3 / [(4n + 3)(2n + 1)!] n=0

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