作者plover (>//////<)
看板Math
标题Re: [分析] 初微(60)
时间Sun Aug 21 09:39:07 2005
※ 引述《Dirichlet ( )》之铭言：
: ※ 引述《plover (>//////<)》之铭言：
: : If Σa_n converges with a_n > 0 for all n, and {a_n} is a
: : decreasing sequence, show that n a_n → 0 as n → +∞.
: By assumption => for any ε>0, there exists an integer N>0 s.t. n ≧ N
: we have a_(n+1) + ... + a_(n+n) < ε/2
: {a_n} is a decreasing seq. => n[a_(2n)] < a_(n+1) + ... + a_(n+n) < ε/2
: Hence 2n[a_(2n)] < ε whenever n ≧ N
: Similarly, a_(n+1) + ... + a_(n+n+1) < ε/2 whenever n ≧ N' for some N'
: then (n+1)a_(2n+1) < a_(n+1) + ... + a_(n+n+1) < ε/2
: so (2n+1)a_(2n+1) < (2n+2)a_(2n+1) < ε
: Hence n(a_n) < ε whenever n ≧ max{N,N'}
这结论可以推出 Σ1/n 是发散的.
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.218.142
1F：→ eggsu :好推论，这种sense好像很难培养…… 05/05 22:50