Re: [考古] 幫幫忙
※ 引述《Aprilgreen (最後一年囉↗☆)》之銘言：
: find the value of x for which the series
: ∞
: Σ(x-1)^n converges
: n=1
: (A)o<x≦1(B)-2≦x<0 (C)-2<x≦0 (D)0≦x<2
: (E)-2<x<0 (F)o<X<2 (G)0≦x≦2 (H)-2≦X≦0
by ratio test
(x-1)^(n+1)
lim ------------- = lim (x-1) = │x-1│ < 1
n→∞ (x-1)^n n→∞
固 0 ＜ x ＜ 2 再考慮端點。
當 x = 0 時 原級數為 Σ(-1)^n 因An ≠ 0 (當n→∞，故div)
當 x = 2 時 原級數為 Σ( 1)^n 因An ≠ 0 (當n→∞，故div)
所以0 ＜ x ＜ 2 即為收斂區間
: 第二題
: from the following equation,use implicit differentiation to fine dy/dx│x=0
: sin(xy^2)=2x+y-1
法一：二邊同時微分
則 cos(xy^2)*(y^2 + 2xy*y`) = 2 + y` 移項後
2 - cos(xy^2)y^2
y` = ------------------
cos(xy^2)*2xy - 1
法二：F(x,y) = 0 dy/dx = -Fx/Fy
F(x,y) = sin(xy^2) -2x -y +1 =0
Fx cos(xy^2)y^2 - 2 2 - cos(xy^2)y^2
y` = - ---- = - ------------------- = ------------------
Fy cos(xy^2)2xy - 1 cos(xy^2)2xy - 1
再將x= 0代入，即得解答。(當x=0時，y=1)
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