Binomial .....

(1+x)^{2010}= C_{0 }+ C_{1}x +C_{2}x^{2} +....... + C_{2010}x^{2010}

integrating both sides,

\frac{(1+x)^{2011} -1}{2011}= C_{0}x+ C_{1}x^{2}/2 +C_{2}x^{3}/3 +....... + C_{2010}x^{2011}/2011

putting x=1/2 and -1/2 ,

\frac{(1+1/2)^{2011} -1}{2011}= C_{0}(1/2)+ C_{1}(1/2)^{2}/2 +C_{2}(1/2)^{3}/3 +....... + C_{2010}(1/2)^{2011}/2011
(eqn 1)

\frac{(1-1/2)^{2011} -1}{2011}= C_{0}(-1/2)+ C_{1}(-1/2)^{2}/2 +C_{2}(-1/2)^{3}/3 +..... + C_{2010}(-1/2)^{2011}/2011
(eqn2)

now (eqn 1) - (eqn2) ,

\frac{(3/2)^{2011} -(1/2)^{2011}}{2011}= 2[C_{0}(1/2)+C_{2}(1/2)^{3}/3 +....... + C_{2010}(1/2)^{2011}/2011]

\frac{(3/2)^{2011} -(1/2)^{2011}}{2011}= [C_{0}/(1.1)+C_{2}/(3.4) +....... + C_{2010}/(2011.4^{1005})]