[試題] 99下 薛克民 微積分甲下 第二次小考

[rod24574575 (天然呆)]

課程名稱︰微積分甲下
課程性質︰必修
課程教師︰薛克民
開課學院：電資學院、工學院、管理學院
開課系所︰電機系、資工系、材料系、資管系
考試日期（年月日）︰100/3/7
考試時限（分鐘）：40分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

Write down all the details or you may not get the full grades.
There could be some problems on the back of the paper.


                                      x  sin(t)
A. (a) Find Taylor series for f(x) = ∫ ──── dt and determine the
                                      0    t
       interval of convergence. (5%)

                  x             ∞  (-1)^(n-1)
   (b) Show that ∫ x^(x) dx =  Σ ────── . Hint: Integrate a series
                  0            n=1     n^(n)
       term by term. (5%)


B. Find the sum of

        ∞    n                         ∞  n^(2)
   (a)  Σ ────(5%)            (b)  Σ ────(5%)
       n=1  2^(n)                      n=1  2^(n)

                                             ∞
   Hint: Starting with the geometric series  Σ x^(n)
                                            n=0


C. (a) Find the Taylor series for ㏑(1+x) (5%)

                                   (1+x)
   (b) Find the Taylor series for e      (5%)

              (1+x)       x
       Hint: e      = e．e

                                        1  x                1
   (c) Find the Taylor series for (1 +──)  in porwers of ──to degree
                                        x                   x
       three. (5%)

       Hint: Consider it in exponential form and use the rusults of (a), (b).

                                  ╭  e       2┌       1  x     ┐   11    ╮
   (d) Evaluate the limit:  lim  x│──*x + x │ (1 +──)  - e │- ──*e │
                           x→∞  ╰  2        └       x        ┘   24    ╯
       (5%)



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※ 編輯: rod24574575     來自: 218.167.197.82       (04/03 16:53)