Prove that `1-^n C_1(1+x)/(1+n x)+^n C_2(1+2x)/((1+n x)^2)-^n C_3(1+3x)/((1+n x)^3)+.`
.`.....(n+1)terms = 0`

Explanation

`S = 1 - .^(n)C_(1)((1+x)/(1+nx))+.^(n)C_(2)(1+2x)/((1+nx)^(2))+"...."`
`=underset(r=0)overset(n)sum(-1)^(r).^(n)C_(r)((1+rx))/((1+nx)^(r))`
`=underset(r=0)overset(n)sum(-1)^(r)[(.^(n)C_(r))/((1+nx)^(r))+(.^(n)C_(r)rx)/((1+nx)^(r))]`
`= underset(r=0)overset(n)sum.^(n)C_(r)(-(1)/(1+nx))^(r)+xunderset(r=0)overset(n)sum(n..^(n-1)C_(r-1))/((1+nx)^(r)) (-1)^(r)`
`= [1-1/(1+nx)]^(n)-((nx)/(1+nx))underset(r=0)overset(n)sum.^(n-1)C_(r-1)(-(1)/(1+nx))^(r-1)`
`=[1-(1)/(1+nx)]^(n)-((nx)/(1+nx))[1-(1)/(nx)]^(n-1)`
`= [1-(1)/(1+nx)]^(n-1)[1-(1)/(1+nx)-(nx)/(1+x)]=0`