1/2!+2/3!+3/4!+……+n/(n+1)!
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1/2!+2/3!+3/4!+……+n/(n+1)!
套公式:n/(n+1)!=1/n!-1/(n+1)!
原式=1-1/2!+1/2!-1/3!+…+1/n!-1/(n+1)!
=1-1/(n+1)!
...........1/(n+1)*(n-1)!
这样做:
原式=1/2!+2/3!+3/4!+……+n/(n+1)!
=(1/2!+1/2!)+(2/3!+1/3!)+(3/4!+1/4!)+......+(n/(n+1)!+1/(n+1)!)-(1/2!+1/3!+1/4!+......+1/n!+1/(n+1)!)
=(1+3/3!+4/4!+......+n/n!+(n+1)/(n+1)!)-(1...
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这样做:
原式=1/2!+2/3!+3/4!+……+n/(n+1)!
=(1/2!+1/2!)+(2/3!+1/3!)+(3/4!+1/4!)+......+(n/(n+1)!+1/(n+1)!)-(1/2!+1/3!+1/4!+......+1/n!+1/(n+1)!)
=(1+3/3!+4/4!+......+n/n!+(n+1)/(n+1)!)-(1/2!+1/3!+1/4!+......+1/n!+1/(n+1)!)
=(1+1/2!+1/3!+1/4!+......+1/n!)-(1/2!+1/3!+1/4!+......+1/n!+1/(n+1)!)
=1-1/(n+1)!
思路就是,在原式每项的后边加上相应阶乘的倒数,然后再在最后减去相应阶乘的倒数,会发现两两抵消,到最后只剩下排头的1和最后的1/(n+1)!
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