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

Prove that .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)

If n >2, then prove that C_1(a-1)-C_2xx(a-2)++(-1)^(n-1)C_n(a-n)=a ,w h e r eC_r=^n C_rdot

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

Prove that (^n C_0)/1+(^n C_2)/3+(^n C_4)/5+(^n C_6)/7+.....+dot=(2^n)/(n+1)dot

Prove that ""^(n)C_r + ""^(n)C_(r-1) = ""^(n+1)C_r

Prove that ^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot

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

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Prove that C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)