Re: [微積] 用柯西證收斂

[JohnMash (Paul)]

※ 引述《gotodmcyo (小情)》之銘言：
: 設{An}為一實數數列滿足
: A1=1  , An+1=1/(1+An)   n=1,2,3.....
: 這題應該是要用柯西的方現來證收斂的八？？
: n無限大時任意兩項的差會趨近於零
: 但不知道要怎麼證才有辦法證出趨近於零？？
: thx!~

          Let a_n + a_{n+1} = a_{n+2},  n=0,1,2,3,....
          and a0=1, a1=1

  then

 a0=1, a1=1, a2=2, a3=3, a4=5, a5=8, a6=13, a7=21,....

  then

          A_n = a_{n-1}/a_n

  we can easily prove that

          a_k * a_{k+3} - a_{k+1} * a_{k+2} =  1  for k is even
                                              -1  for k is odd
   and
          (a_k)^2 - a_{k-1} * a_{k+1} = 1 for k is even
                                       -1 for k is odd


   then

   A1 > A3 > A5 > ....> 0

   A2 < A4 < A6 < ....< 1

   then

      A_k →α  for k is even
          →β  for k is odd

   however,

   |A_n - A_{n+1}| =  1/(a_n * a_{n+1}) →0

   hence, α=β



--

  新梗題  good question

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