Prove that `(C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot`

Prove that 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) .

Prove that : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

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 (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + (C_1)/(2) + (C_2)/(3) + ……. + (C_n)/(n+1) = (2^(n+1) -1)/(n+1)

If C_(0), C_(1), C_(2),..., C_(n) are binomial coefficients in the expansion of (1 + x)^(n), then the value of C_(0) - (C_(1))/(2) + (C_(2))/(3) - (C_(3))/(4) +...+ (-1)^(n) (C_(n))/(n+1) is

Prove that .^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3) if n gt 7 .

Prove that 3C_(1)+7C_(2)+11C_(3)+.........+(4n-1)C_(n)=1+(2n-1)*2^(n)