Prove that `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))`

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))

Prove that sum_(r=0)^(n)r(n-r)C_(r)^(2)=n^(2)(^(2n-2)C_(n))

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 ]

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

Prove that (3!)/(2(n+3))=sum_(r=0)^(n)(-1)^(r)((^nC_(r))/(r+3C_(r)))

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=1)^(m-1)(2r^(2)-r(m-2)+1)/((m-r)^(m)C_(r))=-(1)/(m)

Prove that sum_(r=1)^(n)(-1)^(r-1)(1+(1)/(2)+(1)/(3)+...+(1)/(r))^(n)C_(r)=(1)/(n)