Math 409 Advanced Calculus IFinal ExaminationFall 20041. State the following three theorems.(a) the Bolzano-Weierstrass theorem (about sequences)(b) the intermediate value theorem (about continuous functions)(c) the mean-value theorem (about differentiable functions)2. Define the following three notions.(a) compact interval(b) Cauchy sequence(c) Riemann sum3. (a) State the definition of “the sequence {an}∞n=1converges to thelimit L” in the form “for every ² > 0 . . . ”.(b) Prove from the definition that the sequence½12n¾∞n=1convergesto the limit 0.4. (a) State the definition of “the series∞Xn=1anconverges to the sum S”.(b) Prove from the definition that the series∞Xn=112nconverges to thesum 1.5. (a) State the definition of “the function f is continuous at the point x0”in the form “for every ² > 0 . . . ”.(b) Prove from the definition that the function f (x) = x2is continuousat every point x0.December 14, 2004 Page 1 of 2 Dr. BoasMath 409 Advanced Calculus IFinal ExaminationFall 20046. (a) State the definition of “the function f is differentiable at thepoint a”.(b) Prove from the definition that the function f(x) = x2is differen-tiable at every point a.7. (a) State the definition of “the function f is integrable (in the sense ofRiemann) on the interval [a, b]” in terms of upper sums and lowersums.(b) Prove from the definition that the function f(x) = x2is integrableon the interval [0, 1].8. This question concerns the power series∞Xn=1xnn2. In answering this ques-tion, you may cite relevant theorems from the course.(a) Show that the series converges for every x in the closed interval[−1, 1].(b) Does the series represent an integrable function on the closed in-terval [−1, 1]? Explain why or why not.December 14, 2004 Page 2 of 2 Dr.
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