[閒聊]從龍騰版數甲下的夾擠原理說起
由 亞斯 於 星期二 二月 06, 2007 8:39 pm
#ed_op#DIV#ed_cl#不知道該擺在哪兒#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#就落腳在此#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#放錯請勿怪#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#找極限時用夾擠是常有的事#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#這東西其實很直觀#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#跟高一的學生講時大都能接受(不見得會應用)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#怎麼證我也會#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#只是要用到解析性定義#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#DIV#ed_cl#課本裡要把證明列出我也不反對#ed_op#/DIV#ed_cl#反正不可能考高中生#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#只是......#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#且看龍騰課本的證明#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#[夾擠原理]#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#設{a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#},{b#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#}與{c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#}是三數列,k為一非負整數,而且對任意大於k的正整數n,恆有#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#a#ed_op#SUB#ed_cl#n≤#ed_op#/SUB#ed_cl#b#ed_op#SUB#ed_cl#n≤#ed_op#/SUB#ed_cl#c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl# . 若{a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#}與{c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#}皆收斂且極限值同為L,則數列{b#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#}亦收斂,而且#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#lim(n-->∞)} b#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#=L#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#[證明]#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#當n>k時, a#ed_op#SUB#ed_cl#n≤#ed_op#/SUB#ed_cl#b#ed_op#SUB#ed_cl#n≤#ed_op#/SUB#ed_cl#c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl# ,故有0#ed_op#SUB#ed_cl#≤#ed_op#/SUB#ed_cl#b#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#-a#ed_op#SUB#ed_cl#n≤#ed_op#/SUB#ed_cl#c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#-a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#lim(n-->∞)}(c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#-a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl# )=0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#亦即c#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#-a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#隨著n越來越大而趨近於0,因此b#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#-a#ed_op#SUB#ed_cl#n#ed_op#/SUB#ed_cl#亦隨著n越來越大而趨近於0#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#................#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#看出問題了吧#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#這根本是循環論證#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#難道以前沒人發現嗎?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#真不知該如何教#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#FONT color=#ff0000#ed_cl#感謝yll魔王#ed_op#/FONT#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#發現更可怕的事#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#徐式的參考書也是這樣寫#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#不知其他版本是如何?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#難道是從75學年開始的統編版高三理科數學就是如此嗎?#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#這麼直觀的一個定理#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#不能證就不要證嗎#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#何必畫蛇添足(對不起用了成語)弄了個邏輯有問題的證明#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#寫出來的人恐怕還沾沾自喜#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#自Weierstrass提出"epsilon-delta"的解析性定義之後#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#評價不一#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#但總是解決"消逝的幽靈"的問題#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#而且現在數學系學生都要接觸#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#當然不是數學系的不必鑽研#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#但是這是否意味在高中階段#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#我只要教會學生如何去算就好了#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#記得在第一次大一修微積分時#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#陳金次教授語重心長地表示#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#如果不懂微積分基本定理而只會計算#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#怎麼叫做會微積分?#ed_op#/DIV#ed_cl#