課程名稱︰微積分乙下 課程性質︰必修 課程教師︰張志中 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2014.03.25 考試時限（分鐘）：10:30~12:25 (115分鐘) 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 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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