高中排列数与排列证明化简:1/2!+2/3!+3/4!+……n-1/n!(n属于N*,n≥2)求证:(n+1)!/k!-
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高中排列数与排列证明
化简:1/2!+2/3!+3/4!+……n-1/n!(n属于N*,n≥2)
求证:(n+1)!/k!- /(k -1)!=[(n-k+1)!x ]/k!(k≤n)
化简:1/2!+2/3!+3/4!+……n-1/n!(n属于N*,n≥2)
求证:(n+1)!/k!- /(k -1)!=[(n-k+1)!x ]/k!(k≤n)
![高中排列数与排列证明化简:1/2!+2/3!+3/4!+……n-1/n!(n属于N*,n≥2)求证:(n+1)!/k!-](/uploads/image/z/19932544-64-4.jpg?t=%E9%AB%98%E4%B8%AD%E6%8E%92%E5%88%97%E6%95%B0%E4%B8%8E%E6%8E%92%E5%88%97%E8%AF%81%E6%98%8E%E5%8C%96%E7%AE%80%EF%BC%9A1%2F2%21%2B2%2F3%21%2B3%2F4%21%2B%E2%80%A6%E2%80%A6n-1%2Fn%21%EF%BC%88n%E5%B1%9E%E4%BA%8EN%2A%2Cn%E2%89%A52%EF%BC%89%E6%B1%82%E8%AF%81%EF%BC%9A%28n%2B1%29%21%2Fk%21-)
【1】
注意下:(n-1)/(n!)=[n/n!]-[1/n!]=[1/(n-1)!]-[1/(n!)],则:
S=[(1/1!)-(1/2!)]+[(1/2!)-(1/3!)]+[(1/3!)-(1/4!)+…+[1/(n-1)!-1/(n!)]
=(1/1!)-1/(n!)
【2】
[(n+1)!/k!]-[(n!)/(k-1)!]
=[(n+1)×n!]/[k×(k-1)!]-[(n!)/(k-1)!]
={[(n+1)/(k)]-1}×[(n!)/(k-1)!]
=[(n-k+1)/(k)]×[(n!)/(k-1)!]
={(n-k+1)×(n!)}/[k×(k-1)!]
=(n-k+1)×[(n!)/(k!)]
注意下:(n-1)/(n!)=[n/n!]-[1/n!]=[1/(n-1)!]-[1/(n!)],则:
S=[(1/1!)-(1/2!)]+[(1/2!)-(1/3!)]+[(1/3!)-(1/4!)+…+[1/(n-1)!-1/(n!)]
=(1/1!)-1/(n!)
【2】
[(n+1)!/k!]-[(n!)/(k-1)!]
=[(n+1)×n!]/[k×(k-1)!]-[(n!)/(k-1)!]
={[(n+1)/(k)]-1}×[(n!)/(k-1)!]
=[(n-k+1)/(k)]×[(n!)/(k-1)!]
={(n-k+1)×(n!)}/[k×(k-1)!]
=(n-k+1)×[(n!)/(k!)]
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