推 ej001 :great 12/14 20:47
※ 引述《tasukuchiyan (Tasuku)》之銘言：
: 4.(b)
: Let a(n)>0, a(n+1)/a(n)<=(1-2/n) for n>=3.
: Show that the series of a(n) is convergent.
(n＋1)^(-2)
Since 1 - 2/n ≦ ------------ by 1 - 2x ≦ (1＋x)^(-2) for positive x,
n^(-2)
the sequence {a_n/n^(-2)} is decreasing. So, it is bounded above by C.
It is clear that
Σ a_n converges
since Σ n^(-2) converges.
NOTE. The number "2" can be replaced by any p＞1.
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※ 編輯: math1209 來自: 140.113.25.169 (12/14 20:41)