Question

If ( 1 + x ) ^n = c 0 + c 1 x + c 2 x 2 + ⋯ + c n x n then show that C0/1+C2/3+C4/5+........=2^n/(n+1)?

Answer 1

See the explanation below

Explanation:

`(1+x)^n=c_0+c_1x+c_2x^2+.....+c_nx^n`

Integrating both sides

`int(1+x)^ndx=int(c_0+c_1x+c_2x^2+.....+c_nx^n)dx`

`(1+x)^(n+1)/(n+1)=(c_0)/1x+(c_1x^2)/2+(c_2x^3)/3+........(c_nx^(n+1))/(n+1)`

Let `x=1`

`(1+1)^(n+1)/(n+1)=c_0/1+(c_1)/2+c_2/3+........(c_n)/(n+1)`

`(2)^(n+1)/(n+1)=c_0/1+(c_1)/2+c_2/3+........(c_n)/(n+1)`