已知数列{an},{an}=(n+1)/(n2(n+2)2).则Sn=

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/30 15:45:39

已知数列{an},{an}=(n+1)/(n2(n+2)2).则Sn=
已知数列{an},{an}=(n+1)/(n2(n+2)2).则Sn=

已知数列{an},{an}=(n+1)/(n2(n+2)2).则Sn=
an=(n+1)/[n²(n+2)²]
=(1/4)(4n+4)/[n²(n+2)²]
=(1/4)[(n+2)²-n²]/[n²(n+2)²]
=(1/4)[1/n²- 1/(n+2)²]
Sn=a1+a2+...+an
=(1/4)[1/1²-1/3²+1/2²-1/4²+...+1/n²-1/(n+2)²]
=(1/4)[(1/1²+1/2²+...+1/n²)-(1/3²+1/4²+...+1/(n+2)²)]
=(1/4)[1/1²+1/2²-1/(n+1)²-1/(n+2)²]
=5/16 -1/[4(n+1)²] -1/[4(n+2)²]