an=(2n+1)乘1/3n,求sn
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解决时间 2021-11-09 08:21
- 提问者网友：回忆在搜索
- 2021-11-08 22:06
an=(2n+1)乘1/3n,求sn
最佳答案
- 五星知识达人网友：迟山
- 2021-11-08 23:04
let
S= 1.(1/3)^1 +2.(1/3)^2+...+n.(1/3)^n (1)
(1/3)S= 1.(1/3)^2 +2.(1/3)^3+...+n.(1/3)^(n+1) (2)
(1)-(2)
(2/3)S = [ (1/3)^1 +(1/3)^2 +....+(1/3)^n ] -n.(1/3)^(n+1)
= (1/2)[ 1- (1/3)^n ] -n.(1/3)^(n+1)
2S = (3/2)[ 1- (1/3)^n ] -n.(1/3)^n
an = (2n+1)(1/3)^n
= 2[ n.(1/3)^n ] + (1/3)^n
Sn = a1+a2+...+an
= 2S + (1/2)[ 1- (1/3)^n ]
=(3/2)[ 1- (1/3)^n ] -n.(1/3)^n + (1/2)[ 1- (1/3)^n ]
=2[ 1- (1/3)^n ] -n.(1/3)^n
= 2 - (n+2).(1/3)^n
S= 1.(1/3)^1 +2.(1/3)^2+...+n.(1/3)^n (1)
(1/3)S= 1.(1/3)^2 +2.(1/3)^3+...+n.(1/3)^(n+1) (2)
(1)-(2)
(2/3)S = [ (1/3)^1 +(1/3)^2 +....+(1/3)^n ] -n.(1/3)^(n+1)
= (1/2)[ 1- (1/3)^n ] -n.(1/3)^(n+1)
2S = (3/2)[ 1- (1/3)^n ] -n.(1/3)^n
an = (2n+1)(1/3)^n
= 2[ n.(1/3)^n ] + (1/3)^n
Sn = a1+a2+...+an
= 2S + (1/2)[ 1- (1/3)^n ]
=(3/2)[ 1- (1/3)^n ] -n.(1/3)^n + (1/2)[ 1- (1/3)^n ]
=2[ 1- (1/3)^n ] -n.(1/3)^n
= 2 - (n+2).(1/3)^n
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