15.試求無窮級數=____
三、計算題：每題5分（計算過程要寫清楚）1若= ?
2.四正數a，b，c，d成等比數列，若+ 6 = 8，c + d=162，則公比r = ?
3.—皮球自高80公尺處落下，每次反跳高度爲落下高度的，則此球自落下到靜止共經過多少距離？
2. Determine whetheris a tautology by constructing a truth table. (l0pts)
3. Prove by induction:for all positive integer n.(l0pts)
4. Define S = {a, b, c, d, e).(20 pts)【題組】(a) How many permutations are there to arrange a〜e in a ro
【題組】(b) find the number of the onto functions/from S to {1,2, 3} such that f(a) =1.
5. Define A= {-7, -6, -5,…，6, 7} and a relation R on A such that(10 + 6 pts)【題組】(a)Show the relation
【題組】(b) indicate the corresponding partition on A.
6. Consider the graph on the right, answer the following.Indicate the details of every step of the a
【題組】(b) Use Kruskal’s algorithm to find a minimum spanning tree.(必須標示出邊被選擇的順序）
1. Solve(sin x cos y)dx + (cos x sin y)dy = 0. (20%)
2. Solve y''3y'- 4y=0, y(0) = 2, y'(0) = 1.(20%)
3. Solve by Cramer,srule. (20%)
4. Solveby considering it as eigenvalue problem. (20%)
5. Find the Fourier sine transform and Fourier cosine transform of e-ax ，a＞0. (20%)
1.【題組】
【題組】(b)求 yp(t、(particular solution)
【題組】(c)若起始條件:y(0) = 2, y'(0) = 0，求完整解
4.(a) Find the eigenvalues and eigenvectors of the following matrix.
(b) Find the inverse matrix of
5.【題組】(a)求，c 為逆時針之圓(x2+y2 =9)路徑
【題組】(b)求▽ x(15%)
根據DSM-IV-TR 診斷準則手冊，診斷自閉症時初發年齡，必須在幾歲前？(A) 一歲 (B) 三歲 (C) 七歲 (D) 十歲。