课程名称︰微积分乙下 课程性质︰必修 课程教师︰张志中 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2014.03.25 考试时限（分钟）：10:30~12:25 (115分钟) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. (10%) ∞ n Suppose that Σ a (-3) is convergent. Determine the convergence/divergence n=1 n of the following two series. Explain carefully and give examples if necessary. ∞ n (a) Σ a 2 n=1 n ∞ n (b) Σ a 3 n=1 n 2. (15%) -1 (a) Derive a power series representation for tan x centered at 0. (Find the general term and the radius of convergence R) -1 2 3 (ln(1+x))(tan x)-x + 1/2 x (b) Evaluate ----------------------------- = ? 3 (sinx )(1-cosx) 3. (25%) 1/2 3 Use series to approximate ∫ sin(x )dx so that the magnitude of the error -8 0 is less than 10 . 1/2 3 -8 Ans. ∫ sin(x )dx ≒ ? with |error|≦ ? which is less than 10 . 0 4. (25%) 1+x (a) Derive a power series representation for ln(---) centered at 0. 1-x (Find the general term and the radius of convergence R). (b) Evaluate the first three nonzero terms of the power series to approximate ln 2, and estimate the error. Ans. ln 2 ≒ ? with |error|≦ ? 5. (25%) Consider the function e^x. (a) Write down the Maclaurin series of e^x and its radius of convergence R. The general form of the nth term must be given. No proof is needed. (b) Let T_n(x) be the nth-order Taylor polynomial of e^x at 0 and R_n(x) be the remainder. State the Taylor's inequality for |x|≦d, d>0. No proof is needed. (c) Evaluate the first four nonzero terms of the Maclaurin series to estimate e^0.1. Then show that the magnitude of the error is less than 10^-5. (Hint. Show that e^0.1 < 2 by comparing e with 2^10). Ans. e^0.1 ≒ ? with |error|≦ ? which is less than 10^-5. --

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