Prove that `.^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1)`.

Explanation

` because (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) dx " " [((1=x)^(n+1))/(n+1)]_(0)^(1)`
` rArr = [C_(0) x + (C_(1) x^(2))/(2) + (C_(2) x^(2))/(3) + ...+ (C_(n) x^(n+1))/(n+1)]_(0)^(1)`
` rArr C_(0) + (C_(1))/(2) + (C_(2))/(3) + ...+ (C_(n))/(n+1) = (2^(n+1)-1)/(n+1)`
or ` C_(0) + (C_(1))/(2) + (C_(2))/(3) + ...+ (C_(n))/(n+1) = (2^(n+1) -1)/(n+1)`
I .Aliter
` LHS = C_(0) + (C_(1))/(2) + (C_(2))/(3) + ...+ (C_(n))/(n+1)`
` = 1 + (n)/(1*2) + (n(n-1))/(1*2*3) + ...+ (1)/(n+1)`
` = (1)/(n+1) [(n+1) + ((n+1)n)/(1*2) + ((n+1)n(n-1))/(1*2*3) + ...+ 1]`
Put n + 1 = N , then
`LHS = (1)/(N) [N+ (N(N-1))/(2!) + (N(N-1)(N-2))/(1*2*3) + ...+ 1]`
` = (1)/(N) [ ""^(N)C_(1) + ""^(N)C_(2) + ""^(N)C_(3) + ...+ ""^(N)C_(N)]`
` = (1)/(N) [ (1 + 1)^(N)-1] = (2^(N)-1)/(N) = (2^(n+1) -1)/(n+1) = RHS`
II. Aliter
` LHS = C_(0) + (C_(1))/(2) + (C_(2))/(3) + ...+ (C_(n))/(n+1) = sum_(r=0)^(n) (C_(r))/(r+1)`
` = sum_(r=0)^(n)(""^(n)C_(r))/((r+1)) = sum_(r=0)^(n) (""^(n+1)C_(r+1))/((n+1)) [ because (""^(n+1)C_(r+1))/(n+1) = (""^(n)C_(r))/(r+1)]`
` = (1)/((n+1)) sum_(r=0)^(n) ""^(n+1)C_(r+1)`
` (1)/((n+1)) (""^(n+1)C_(1) + ""^(n+1)C_(2) + ""^(n+1)C_(3) + ...+""^(n+1)C_(n+1) `
` = (1)/(n+1) (2^(n+1)-1) = (n^(n+1) -1)/(n+1) = RHS ` .