[微積] 台大數研90高微考古題

[tasukuchiyan (Tasuku)]

4.(b)
Let a(n)>0, a(n+1)/a(n)<=(1-2/n) for n>=3.
Show that the series of a(n) is convergent.

6.
Prove that, for each integer n, there exists a C^2 function w=g(x,y)
defined in some neighborhood of (0,0) such that
x+2yw+cosw=0,
g(0,0)=nπ+π/2.

請問有任何解題的方向嗎?

謝謝

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◆ From: 134.208.26.13
推 bineapple   :6 implicit function theorem
→ tasukuchiyan:隱函數定理我有想過，不過不知道要怎麼使用
→ tasukuchiyan:嚴格的寫下證明
→ Sfly        :4的不等號反了吧
→ bineapple   :F(x,y,w)=x+2yw+cosw=0 at (0,0,nπ+π/2)
→ bineapple   :所以可以把w寫成g(x,y)在一個neighborhood of (0,0)
→ bineapple   :然後對F(x,y,g(x,y))偏微 證明g是C^2
※ 編輯: tasukuchiyan    來自: 134.208.26.13        (12/11 17:40)
→ tasukuchiyan:要怎麼對F(x,y,g(x,y))偏微，證明g是C^2呢?
→ bineapple   :F(x,y,g(x,y))在(0,0)的附近都是0 所以得到在0的附近
→ bineapple   :x+2yg(x,y)+cos(g(x,y))=0 對這個式子偏微後能夠把
→ bineapple   :g的一次微分提出來寫成x,y,g(x,y)的式子 然後再利用
→ bineapple   :g是C^1的性質
→ tasukuchiyan:那g'不是從(0,0)的鄰域映射到L(R^2,R)的函數嗎?
→ tasukuchiyan:怎麼用g是C^1的性質去證明g'可微分?
→ Sfly        :把w_x表成 w跟y,x的函數
→ tasukuchiyan:If all mixed second order partial derivatives are
→ tasukuchiyan:continuous at a point, f is termed a C^2 function
→ tasukuchiyan:at that point.
→ tasukuchiyan:是這一回事嗎?
→ bineapple   :yes
→ Sfly        :這句話要證明  雖然只是隱函數定理的一個推論
→ tasukuchiyan:感謝所有回答的人