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MATH 138-1065-Final2_exam

MATH 138-1065-Final2_exam - MATH 138 FINAL EXAM Spring Term...

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Unformatted text preview: MATH 138 - FINAL EXAM. Spring Term 2006 Page 8 of 15 7. Consider the power series n=2 a) Find the radius of convergence of this power series. b) By checking the convergence of the given series at each end point, identify the precise interval of convergence of this power series. MATH 138 - FINAL EXAM. Spring Term 2006 Page 9 of 15 8. Let 2:11an — 2)" be a power series. Suppose the series converges at :1: = —2 and diverges at a: = —3. a) For each of the following points, state whether the series “converges”, “diverges”, or “could either converge or diverge” by checking the appropriate box next to each :6 value. COULD EITHER CONVERGE CONVERGES DIVERGES ' OR DIVERGE b) For the coeflicents on from part a), explain why the series 2:161; and 2:; non must both converge. MATH 138 - FINAL EXAM. Spring Term 2006 Page 10 of 15 9. Below is a graph of a twice differentiable function f (11:). The second degree Taylor polynomial for f at :5 = 0 is of the form co + c123 + 62x2. Deter- mine the signs (positive, negative, or zero) of co, cl, and c2. Explain your reasoning briefly. MATH 138 - FINAL EXAM. Spring Term 2006 Page 11 of 15 10. Let Tn(:c) be the nth Taylor polynomial of e: centered at a: = 0. Using Taylor’s inequality, find the least integer n such that the error Ie — T n(1)| is less than or equal to fl. MATH 138 - FINAL EXAM. Spring Term 2006 Page 12 of 15 11. 3.) Find the Taylor series for f e‘zzdz centered at a: = 0 and state its radius of conver- gence. b) Using the Taylor series for f e‘zzdx from part a), write the number fol 6‘12dx as the sum of a convergent series. G) Let T9(z) be the 9th degree Taylor poynomial of f eflzdx at z = 0. Using the Al- ternating Series error bound, estimate the error involved in approximating fol e‘fldx MATH 138 - FINAL EXAM. Spring Term 2006 Page 13 of 15 12. Label each statement as TRUE or FALSE in the blank provided. Give brief justifications for true answers, and counter examples for false answers. Unjustified answers will not be graded. a) The series 23:1 5—713 converges conditionally b) If limn_.o<,(an+1 + (1,.) = 0, then 2:11 a" converges. c) If the Taylor series of f (z) centered at a: = O diverges at a: = ——1, then f (z) is not defined at z : —1. d) The series 22:0 3333 converges to the value 65‘”. e) If the Taylor series for f (x) is 2:11 aux", and the Taylor series for 9(1‘) is 2:11 bar“, then the Taylor series for f (:c)g(:v) is Z;l(anb,.)x". ...
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