数列2/2,4/2^2,6/2^3,……,2n/2^n,……的前n项的和sn=
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![数列2/2,4/2^2,6/2^3,……,2n/2^n,……的前n项的和sn=](/uploads/image/z/207420-60-0.jpg?t=%E6%95%B0%E5%88%972%2F2%2C4%2F2%5E2%2C6%2F2%5E3%2C%E2%80%A6%E2%80%A6%2C2n%2F2%5En%2C%E2%80%A6%E2%80%A6%E7%9A%84%E5%89%8Dn%E9%A1%B9%E7%9A%84%E5%92%8Csn%3D)
数列2/2,4/2^2,6/2^3,……,2n/2^n,……的前n项的和sn=
数列2/2,4/2^2,6/2^3,……,2n/2^n,……的前n项的和sn=
数列2/2,4/2^2,6/2^3,……,2n/2^n,……的前n项的和sn=
Sn =2/2 + 4/2² + 6/2³ + …… + 2(n-1)/2^(n-1) + 2n/2^n …………①
2Sn=2×(2/2 + 4/2² + 6/2³ + …… + 2n/2^n)
=2 + 4/2 + 6/2² + 8/2³ + …… + 2n/2^(n-1) …………②
②减①得
Sn=2 + 2/2 + 2/2² + 2/2³ + …… +2/2^(n-1) - 2n/2^n
=2[1-(1/2)^n]/[1-(1/2)] - 2n/2^n
=4 - 4/2^n - 2n/2^n
=4 - (2n+4)/2^n
=4 - (n+2)/2^(n-1)
这叫错位相减法