1/2×4+1/3×5+.1/（n+1)(n+3)等于?1+2/3+3/3².+n/3的n-1次方 等于?1+1/1+2+1/1+2+3.+1/1+2+3..+1/1+2+3..+n 等于?

1/2×4+1/3×5+.1/（n+1)(n+3)等于?1+2/3+3/3².+n/3的n-1次方 等于?1+1/1+2+1/1+2+3.+1/1+2+3..+1/1+2+3..+n 等于?
1/2×4+1/3×5+.1/（n+1)(n+3)等于?1+2/3+3/3².+n/3的n-1次方 等于?
1+1/1+2+1/1+2+3.+1/1+2+3..+1/1+2+3..+n 等于?

1/2×4+1/3×5+.1/（n+1)(n+3)等于?1+2/3+3/3².+n/3的n-1次方 等于?1+1/1+2+1/1+2+3.+1/1+2+3..+1/1+2+3..+n 等于?
1/(n+1)(n+3)=[1/(n+1)-1/(n+3)]/2
1/2×4+1/3×5+.1/（n+1)(n+3)
=(1/2)*(1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+...+1/(n+1)-1/(n+3))
=(1/2)*(1/2+1/3-1/(n+2)-1/(n+3))
=(1/2)*(5/6-1/(n+2)-1/(n+3))
Sn= 1+2/3+3/3^2+...+(n-1)/3^(n-2)+n/3^(n-1)
3Sn=3+2+3/3+4/3^2+...+n/3^(n-2)
2Sn=3+[1+1/3+1/3^2+...+1/3^(n-2)]-n/3^(n-1)
=3-n/3^(n-1)+[1-(1/3)^(n-1)]/(1-1/3)
=3-n*3^(1-n)+3/2-(3/2)*3^(1-n)
=9/2-(n+3/2)*3^(1-n)

1/（n+1)(n+3)=<1/（n+1）-1/(n+3)>/2
所以1/2×4+1/3×5+.....1/（n+1)(n+3）
=(1/2-1/4+1/3-1/5......1/(n+1)-1/(n+3))/2
=5/12+(n+2)/(n+1)*(n+3)

1

1+2/3+3/3²......+n/3^(n-1)
构造函数f(x)=1+(x/3)+(x/3)^2+(x/3)^3+……(x/3)^n (1)
对f(x)求和得f(x)=[1-(x/3)^(n+1)]/(1-x/3) (2)
对f(x) (1)求导得f(x)=1/3+2*(x/3)+3*(x/3)^2+……n*(x/3)^(n-1)
所以原式=f(1)-2/3
对2求导可以计算出f(1)再代入可以求出原式的值