数列an=n/2^(n+1)求和再求极限
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数列an=n/2^(n+1)求和再求极限
![数列an=n/2^(n+1)求和再求极限](/uploads/image/z/17740014-6-4.jpg?t=%E6%95%B0%E5%88%97an%3Dn%2F2%5E%EF%BC%88n%2B1%EF%BC%89%E6%B1%82%E5%92%8C%E5%86%8D%E6%B1%82%E6%9E%81%E9%99%90)
Sn=1/2^2+2/2^3+3/2^4+4/2^5+……+(n-1)/2^n+n/2^(n+1)
2Sn=1/2+2/2^2+3/2^3+4/2^4+……+(n-1)/2^(n-1)+n/2^n
两式相减:
Sn=1/2+1/2^2+1/2^3+1/2^4+1/2^5+……+1/2^n-n/2^(n+1)
=(1/2)[(1/2)^n-1]/(1/2-1)-n/2^(n+1)
=1-(1/2)^n-n(1/2)^(n+1)
=1-2(1/2)^(n+1)-n(1/2)^(n+1)
=1-(2+n)(1/2)^(n+1)
limSn=lim[1-(2+n)(1/2)^(n+1)]
=1-lim[(2+n)(1/2)^(n+1)]
=1
2Sn=1/2+2/2^2+3/2^3+4/2^4+……+(n-1)/2^(n-1)+n/2^n
两式相减:
Sn=1/2+1/2^2+1/2^3+1/2^4+1/2^5+……+1/2^n-n/2^(n+1)
=(1/2)[(1/2)^n-1]/(1/2-1)-n/2^(n+1)
=1-(1/2)^n-n(1/2)^(n+1)
=1-2(1/2)^(n+1)-n(1/2)^(n+1)
=1-(2+n)(1/2)^(n+1)
limSn=lim[1-(2+n)(1/2)^(n+1)]
=1-lim[(2+n)(1/2)^(n+1)]
=1