Re: [分析] 初微(60)
※ 引述《Dirichlet ( )》之銘言：
: ※ 引述《plover (>//////<)》之銘言：
: : 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'}
這結論可以推出 Σ1/n 是發散的.
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05/05 22:50, , 1F
05/05 22:50, 1F
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