1/3 .3/9 .5/27 .2n-1/3^n求和

1/3 .3/9 .5/27 .2n-1/3^n求和
1/3 .3/9 .5/27 .2n-1/3^n求和

1/3 .3/9 .5/27 .2n-1/3^n求和
设 x =1/3^1 + 3/3^2 + 5/3^3 +…… + 2n-1/3^n
3x = 1 + 3/3^1 + 5/3^2 +…… + 2n-1/3^(n-1)
3x - x =[ 1 + 3/3^1 + 5/3^2 +…… + 2n-1/3^(n-1) ] - [ 1/3^1 + 3/3^2 + 5/3^3 +…… + 2n-1/3^n ]
即
2x = 1 + 2/3^1 + 2/3^2 +…… + 2/3^(n-1) - 2n-1/3^n
2x = 1 + 2 * [ 1/3^1 + 1/3^2 +…… + 1/3^(n-1) + 1/3^n ] - 2/3^n - 2n-1/3^n
x = [ 1/3^1 + 1/3^2 +…… + 1/3^(n-1) + 1/3^n ] + [ 1/2 - 2n+1/(2 * 3^n) ]
设 y= 1/3^1 + 1/3^2 +…… + 1/3^n
3y= 1 + 1/3^1 + 1/3^2 +…… + 1/3^(n-1)
3y - y = [ 1 + 1/3^1 + 1/3^2 +…… + 1/3^(n-1) ] - [ 1/3^1 + 1/3^2 +…… + 1/3^n ]
2y = 1 - 1/3^n
y = 1/2 - 1/(2 * 3^n)
代入 x 的算式：
x = [ 1/2 - 1/(2 * 3^n) ] + [ 1/2 - 2n+1/(2 * 3^n) ]
x = 1 - 1/(2 * 3^n) - 2n+1/(2 * 3^n)
x = 1 - (2n + 2 )/(2 * 3^n)
x = 1 - (n + 1)/3^n
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