天助
Lv 7
天助 asked in 科學數學 · 1 decade ago

求limit值(2)

已知0<a(1)<1, a(n+1)=arctan[a(n)], n>0, 則 lim(n->∞) (√n * a(n) )=?

Update:

Ans: √(3/2)

4 Answers

Rating
  • 1 decade ago
    Favorite Answer

    性質: lim(n->∞)[a(n+1)-a(n)]=A, 則lim(n->∞) a(n)/n=A

    利用以上性質比較簡單!

    2010-02-20 22:32:34 補充:

    1. 顯然 a(n)遞減至0 (至少n夠大時如此)

    2. 設b(n)= 1/[a(n)]^2, 則

    lim(n->∞) [b(n+1)-b(n)]

    =lim(n->∞)[ 1/arctan^2(a(n)) - 1/a(n)^2]

    (代換x=arctan(a(n)) 得 lim(x->0) (1/x^2 - 1/tan^2(x))

    = lim(x->0)[ (tanx +x)(tanx - x)/(xtanx)^2]

    = 2/3 (以Taylor's expansion計算,請自行計算, OK!?)

    3. 由意見所述性質知 lim(n->∞) b(n)/n = 2/3

    則 lim(n->∞) n*[a(n)]^2 = 3/2

    故原題limit =√(3/2)

  • 天助
    Lv 7
    1 decade ago

    分析能力很強,嚴謹度可再補強嗎?

    2010-02-20 22:13:46 補充:

    有點籠統,請學長講仔細一點,好嗎?

    2010-02-20 22:33:59 補充:

    謝 謝!

  • 1 decade ago

    ====================================================

    圖片參考:http://img163.imageshack.us/img163/1293/79790581.p...

    2010-02-09 20:00:24 補充:

    http://img163.imageshack.us/img163/1293/79790581.p...

  • 大師
    Lv 7
    1 decade ago

    call 高錕啦

    佢真係可以幫到你嫁

    唔啱call 陳振聰啦

    佢乜都識嫁

    山頂可賦山道 24號

Still have questions? Get your answers by asking now.