Re: [分析] 級數

[math99 (math99)]

※ 引述《iamagine (A-gine)》之銘言：
: 這個級數是收斂or發散?
:  ∞
:  Σ  _______1_______
: n=3  (ln n)^(lnln n)
: 麻煩大大們了>""<


Since                                              2
            lnln n       lnln n  lnln n    (lnln n)
      (ln n)         = (e       )       = e          ,

                      2
              (lnln n)
and      lim -----------  = 0    (by L.R.).
               ln n
                                                lnln n
then there is N > 0 suth that n > N implie 0 < --------- < 1.
                                                 ln n

That is,
                      2
          0 < (lnln n)   < ln n        (n > N)
                       2
               (lnln n)
=>        0 < e          < n           (n > N)

              1           1
=>       -----------  > ------         (n > N).
                   2
           (lnln n)       n
           e

                                     1
Hence, by compare test, we have Σ ----- diverge implies
                                     n
               ∞           1
               Σ   ----------------- diverge.
               n=3  (ln n)^(lnln n)


--
小試一下，有錯請指正！^^


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※ 編輯: math99          來自: 123.195.80.144       (04/04 21:00)
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