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1/1*2+1/2*3+1/3*4+...+1/n(n+1)化简

1/n(n+1) = 1/n - 1/(n + 1)1/1*2+1/2*3+1/3*4+…+1/n(n+1)= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + + 1/n - 1/(n + 1)= 1 - 1/(n + 1)= n/(n + 1)

1/(1*2)+1/(2*3)+1/(3*4)+…+1/(n*(n+1))=1-1/2+1/2-1/3……+1/n-1/(n+1)=1-1/(n+1)=n/(n+1)有不懂欢迎追问

原式=1-1/2+1/2-1/3+1/3-1/4++1/n-1/(n+1) =1-1/(n+1) =n/(n+1)

1/[n*(n 1)*(n 2)]=1/2*[1/(n*(n 1))-1/(n 1)*(n 2)]1/[n*(1 n)]=1/n-1/(n 1)1/[(1 n)*(2 n)]=1/(n 1)-1/(2 n)再求和其中很多项都抵消了最后的和为:S=0.25-[1/(n*n)]/[1 (3/n) (2/(n*n))]就是化简后的结果了

1/N(N+1)=1/N-1/(N+1) 1/1*2=1-1/2 1/2*3=1/2-1/3 1/1*2+1/2*3+1/3*4++1/N(N+1)=1-1/(N+1)

1/1*2+1/2*3+1/3*4++1/n(n+1)=1-1/2+1/2-1/3+1/3-1/4…………+1/n-1/(n+1)(中间全部抵消)=1-1/(n+1)=n/(n+1)祝你学习进步!如有疑问请追问,理解请及时采纳!(*^__^*)

根据:1/n(n+1)=1/n-1/(n+1) 1/1*2+1/2*3+1/3*4+..1/n(n+1) =1/1-1/2+1/2-1/3+1/3-1/4++1/(n-1)-1/n +1/n-1/(n+1) =1-1/(n+1) =n/(n+1)

通项=n(n+1)(n+2)=n^3+2n^2+2n再利用公式1+2^3+…+n^3=(1/4)*n^2*(n+1)^2;1+2^2+…+n^2=(1/6)*n*(n+1)*(2n+1);1+2+…+n=(1/2)*n*(n+1);将上面三式相加整理即可

解:原式=(1-1/2)+(1/2-1/3)+(1/3-1/4)+…+[1/n-1/(n+1)]=1-1/2+1/2-1/3+1/3-1/4+……+1/n-1/(n+1)=1-1/(n+1)=n/(n+1)

+1/n(n+1)=1-1/n(n+1)=1-1/n(n+1)=(n^2+n-1)/..+1/n-1/2+1/3+.;2-1/..1/(1*2)+1/(2*3)+1/(3*4).

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