`sum_(r=0)^n (-1)^r C(n,r)((1/(2^r))+((3^r)/(2^(2r)))+((7^r)/(2^(3r)))+...oo` is equal to

Show that sum_(r=0)^(n) (-1)^(r ). ""^(n)C_(r ) {(1)/(2^(r )) +(3^(r ))/(2^(2r)) +(7^(r ))/(2^(3r)) +…."upto m terms"}=(2^(mn)-1)/(2^(mn)(2^(n)-1))

If sum_(r=0)^(n) (-1)^(r ).""^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+"……..""to m terms"] = k(1-1/(2^(mn))) , then k =

find the sum of the series 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 ]

sum_(r=0)^(oo)((r+1)((2)/(3))^(r)) is

sum_(r=0)^n(-1)^r .^n C_r[1/(2^r)+3/(2^(2r))+7/(2^(3r))+(15)/(2^(4r))+ .....mt e r m s]= (2^(m n)-1)/(2^(m n)(2^n-1))

sum_(r=0)^(n)((n-3r+1)^(n)C_(r))/((n-r+1)2^(r)) is equal to

sum_(r=0)^n(-1)^r^n C_r[1/(2^r)+3/(2^(2r))+7/(2^(3r))+(15)/(2^(4r))+ u ptomt e r m s]= (2^(m n)-1)/(2^(m n)(2^n-1))

underset(r=0)overset(n)(sum)(-1)^(r).^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))+ . . .m" terms"]=

underset(r=0)overset(n)(sum)(-1)^(r).^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))+ . . .m" terms"]=