Prove that `C_0+(C_1)/(2)+(C_2)/(3)+....+(C_n)/(n+1)=(2^(n+1)-1)/(n+1)`

e_(0)+(C)/(2)+(C_(2))/(3)++(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3)x^(3) + ...+ C_(n)x^(n) , prove that (C_(1))/(2) + (C_(3))/(4) + (C_(5))/(6) + …= (2^(n+1)-1)/(n+1) .

Prove that (i) C_(1)+2C_(2)+3C_(3)+……+nC_(n)=n.2^(n+1) (ii) C_(0)+(C_(1)/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(n+1) .

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + C_(3) x^(3) + …+ C_(n) x^(n) prove that (C_(0))/(1) + (C_(2))/(3) + (C_(4))/(5) + ...= (2^(n))/(n+1) .

C_(0)-(C_(1))/(2)+(C_(2))/(3)-......+(-1)^(n)(C_(n))/(n+1)=(1)/(n+1)

C0-(C1)/(2)+(C2)/(3)-............+(-1)^(n)(Cn)/(n+1)=(1)/(n+1)

Prove that (C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))(C_(n-1)+C_(n))=((n+1)^(n))/(n!)*c_(0)*C_(1)*C_(2).........