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Find the sum of the series 1+2(1-x)+3(1-...

Find the sum of the series `1+2(1-x)+3(1-x)(1-2x)++n(1-x)(1-2x)(1-3x)[1-(n-1)x]` .

Text Solution

Verified by Experts

The correct Answer is:
`1/x[1-(1-x)(1-2x)...(1-nx)`
`T_(r)=r(1-x)(1-2x)(1-3x)…(1-(r-1)x)`
`=(1-x)(1-2x)(1-3x)…(1-(r-1)x)(xr-1+1)/x`
`=1/x{(1-x)(1-2x)…(1-(r-1)x)-(1-x)(1-2x)..(1-(r-1)x)(1-rx)}`
`=1/x[V(r-1)-V(r )]`, where
`V(r )=(1-x)(1-2x)…(1-rx)`
`therefore` Required sum=`sum_(r=1)^(n)T_(r)=1/xsum_(r=1)^(n)V(r-1)-V( r)`
`=1/x(V(0)-V(n))`
`1/x[1-(1-x)(1-2x)...(1-nx)]`
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