Question

数列n/n+1怎么求和

Answer 1

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

Answer 2

先求1/2+1/3+...+1/(n+1)，再用n来减。