課程名稱：高等微積分(二) 課程性質：必修 課程範圍：課本ch8~ch9 開課教師：蔡紋琦 開課學院：商學院 開課系級：統計二 考試日期(年月日)：2009/04/14 考試時限(Mins)：二234 試題本文： 1.(40 pts) For each of the following statements, determine whether it is true or false. (a)___The Taylor series expansion of the function f:R→R define by f(x) = e^x^2+e^(-x^2) about the point x0 = 0 is 2+2x^4/2!+2x^8/4!+2x^12/6!+... for all x in R. (b)___The equality ln(1+x) = x-x^2/2+x^3/3-x^4/4+...holds only when -1 < x < 1 (c)___If a sequence of functions {|fn|} converges pointwise to a function |f| on D, then {f} also converges pointwise to f on D. (d)___Suppose that the function f:R→R is continuous. Then for each positive number ε, there is a polynomial p:R→R such that |f(x)-p(x)| <ε for all points x in R. ∞ ∞ (e)___Suppose that Σak and Σbk are series of positive numbers such that k=1 k=1 ∞ lim(bk/ak) = l > 0. Then the series Σak diverges if and only if the series k=1 ∞ Σbk diverges. k=1 (f)___Suppose that both the sequences {fn:D→R} and {gn:D→R} are uniformly Cauchy on D. Then the sequence{fn-gn:D→R} is also uniformly Cauchy. (g)___If fn is a sequence of continuous functions on D converging to a continuous function f on D, then the convergence is uniform. (h)___Suppose that {fn:(0,1)→R} is a sequence of continuously differentiable function that converges uniformly to the function f:(0,1)→R. Then the limit function f is also continuously differentiable and fn'(x)→f'(x) for all x in (0,1). ∞ (i)___If the power seriesΣakx^k diverges when x = -1/3, then it also diverges k=0 when x = 2/3. ∞ (j)___Consider the power series Σakx^k. Suppose that lim |ak|^(1/k) = α for k=0 k→∞ ∞ some α> 0. Then the radius of convergence of the seriesΣakx^k is 1/α. k=0 2.(8pts)Please compute the third Taylor polynomial for the function f:R→R x defined by f(x) =∫e^(-t^2) dt at the point x0 = 0. Show your work. 0 3.(8pts)Please state the Cauchy integral remainder theorem. 4.(10pts)Let n be an odd natural number and x0 is a point in R. Suppose that (n+1) f:R→R has n+1 derivatives and that f :R→R is continuous. Assume that (k) (n+1) f (x0) = 0 if 1≦k≦n and that f (x0) > 0. Please use the Lagrange remainder theorem to verify that x0 is a local minimizer. 5.For each number n and each x in[0,1], define fn(x) = 1/(nx+1). (a)(6pts)Find the function f:[0,1]→R to which the sequence{fn:[0,1]→R} converges pointwise. (b)(8pts)Prove the convergence in (a) is not uniform. 6.(10pts)Please give an example of a sequence{fn:[0,1]→R} where the sequence {fn} converges pointwise to f on [0,1] but sup fn(x) does not converge to 0≦x≦1 sup f(x). 0≦x≦1 7.(10pts)For each natural number n, let the function fn:R→R be bounded. Suppose that the sequence{fn} converges uniformly to f on R. Prove that the limit function f:R→R also is bounded. --

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