[" age "-1],[" Suppose "f(n,m)=sum_(r=0)^(n)(-1)^(r)C_(r)((1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+..." up to "m" terms ")]

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 ]

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))

Sigma_(r=0)^(n)(-1)^(r).^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+...up rarr mterms ]

Let for n in N, f(n)=sum_(r=0)^(n)(-1)^(r)(C_(r)2^(r+1))/((r+1)(r+2))

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 =

Prove that sum_(r=0)^(n)(-1)^(r)nC_(r)[(1)/(2^(r))+(3)/(2^(2r))+(7)/(2^(3r))+(15)/(2^(4r))+...up to mterms]=(2^(mn)-1)/(2^(mn)(2^(n)-1))

The value of sum_(r=0)^(3n-1)(-1)^(r)6nC_(2r+1)3^(r) is

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"]=

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