已知数列{a(n)}满足a(1)=3 a(n)=(2n-1)/3^n (n>1)求s(n)
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已知数列{a(n)}满足a(1)=3 a(n)=(2n-1)/3^n (n>1)求s(n)
Sn=a1+a2+a3+a4+.+an
Sn=3 + 3 / 3^2 + 5 / 3^3 + 7 / 3^4 +.+ (2n-1) / 3^n-------(1)
两边同时除以 3 得
1/3 Sn = 1 + 3 / 3^3 + 5 / 3^4 + 7 / 3^5 +.+ (2n-1) / 3^(n+1)----(2)
(1)-(2)得
2/3 Sn = 2 + 3 / 3^2 + 2 / 3^3 + 2 / 3^4 +.+ 2 / 3^n - (2n-1) / 3^(n+1)
= 2 + 1 / 3^2 +2 / 3^2 + 2 / 3^3 + 2 / 3^4 +.+ 2 / 3^n - (2n-1) / 3^(n+1)
= 2 + 1 / 3^2 - 2 / 3^1 + 2 / 3^1+2 / 3^2 + 2 / 3^3 + 2 / 3^4 +.+ 2 / 3^n - (2n-1) / 3^(n+1)
= 13/9 +2 [1 / 3^1 + 1 / 3^2 + 1 / 3^3 + 1 / 3^4 +.+ 1 / 3^n ] - (2n-1) / 3^(n+1)
= 13/9 + 1 - 1 / 3^n - (2n-1) / 3^(n+1)
= 22/9 - (6n-2) / 3^n
Sn = 11/3 - (3n - 1) / 3^(n-1)
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