用裂项相消法做下列3个题 1.an=1/n(n+1) 2.an=1/(3n+1)(3n+2) 3.an=1/n²+2n
问题描述:
用裂项相消法做下列3个题 1.an=1/n(n+1) 2.an=1/(3n+1)(3n+2) 3.an=1/n²+2n
答
an=1/[n(n+1)]=(1/n)- [1/(n+1)](裂项)
则 Sn=1-(1/2)+(1/2)-(1/3)+(1/3)-(1/4)…+(1/n)- [1/(n+1)](裂项求和)
= 1-1/(n+1)
= n/(n+1)
同理
2.an=1/(3n+1)(3n+2)=[1/(3n+1)]- [1/(3n+2)]
3.an=1/n²+2n=1/[n(n+2)]=1/2【(1/n)- [1/(n+2)]】