已知数列1+1/3,2+1/9,3+1/27,...,n+(1/n^3),.求该数列的前n项和Sn
来源:学生作业帮助网 编辑:作业帮 时间:2024/09/01 14:54:24
已知数列1+1/3,2+1/9,3+1/27,...,n+(1/n^3),.求该数列的前n项和Sn
已知数列1+1/3,2+1/9,3+1/27,...,n+(1/n^3),.求该数列的前n项和Sn
已知数列1+1/3,2+1/9,3+1/27,...,n+(1/n^3),.求该数列的前n项和Sn
1+1/3+2+1/9+3+1/27+...+n+(1/n^3)
=1+2+3+……+n+(1/3+1/9+1/27+……+1/n^3)
=n(1+n)/2+(1-1/n^3)/2
数列为:1+1/3,2+1/9,3+1/27,...,n+(1/3^n)
该数列的前n项和:
Sn=(1+1/3)+(2+1/9)+(3+1/27)+...+[n+(1/n^3)]
=(1+2+3+…+n)+(1/3+1/9+1/27+1/3^n)
=n(n+1)/2+1/3(1-1/3^n)/(1-1/3)
=n(n+1)/2+(1-3^n)/2
=[n(n+1)+(1-3^n)]/2
(1+1/3)+(2+1/9)+(3+1/27)+...+【n+(1/3^n)】
=(1+2+3+……+n)+(1/3+1/9+1/27+……+1/3^n)
=n(1+n)/2+(1-1/3^n)/2
=(n² + n + 1 - 1/3^n )/ 2