Prove that
`sum_(r=0)^(n) (-1)^(r)""^(n)C_(r)[1/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))+"......up to m terms "]`
`= (2^(mn)-1)/(2^(mn)(2^(n)-1))`

Explanation

`underset(r=0)overset(n)sum(-1)^(r)xx.^(n)C_(r)(1/(2^(r))+(3/4)^(r)+(7/8)^(r)+".....m terms")`
`= underset(r=0)overset(n)sum.^(n)C_(r)((-1)/(2))^(r)+underset(r=0)overset(n)sum.^(n)C_(r)((-3)/(9))^(n)+underset(r=0)overset(n)sum.^(n)C_(r)((-7)/(8))^(r)+".......m terms"`.
`=(1-(1)/(2))^(n) + (1-3/4)^(n) + (1-7/8)^(n)+(1-15/16)^(n)+"....."m` terms a
`= ((1)/(2^(n))[1-(1/(2^(n)))^(m)])/(1-1/(2^(n)))= (2^(mn)-1)/(2^(mn)(2^(n)-1))`