we have (1+x)ⁿ as = C₀+C₁x+C₂x²+C₃x³.........................C_nxⁿ......................1)

and we have  (1+1/x)ⁿ= C₀+C₁1/x+C₂ 1/x²+ C₃ 1/x³....................C_n 1/xⁿ........2)

now multiplying 1) and 2).................well have the desired series as the coefficient of the constant term...........

=coeff of const term in expansion of (1+x)ⁿ(1+1/x)ⁿ= (1+x)²ⁿ/ xⁿ.....................................3)

= coeff of xⁿ in the expansion of  (1+x)²ⁿ

= ²ⁿC_n

Hence proved.....