X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)

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X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)
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X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)
X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)

X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)
Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)=1/(x1+x2+...+xn-1)-1/(x1+x2+...+xn-1+xn)
所以原式=1/x1-1/(x1+x2)+1/(x1+x2)-1/(x1+x2+x3)+...+1/(x1+x2+...+xn-1)-1/(x1+x2+...+xn-1+xn)=1/x1-1/(x1+x2+...+xn-1+xn)
这叫做裂项求和

X2/X1(X1+X2)+X3/(X1+X2)(X1+X2+X3)+.........Xn/(x1+x2+...Xn-1)(X1+X2...+Xn)