sum_(i=0)^(n)(-1)'*^(n)C_(r)((1)/(2')+(3')/(2^(2))+(7')/(2^(3t))+........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))

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

sum_(0)^(oo)(1)/((3n+1)(3n+2))

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))

The value of lim_(n rarr oo)sum_(r=0)^(n)(sum_(t=0)^(r-1)(1)/(5^(n))*^(n)C_(r)^rC_(t)*(3^(t))) 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))

Lt_(n rarr oo)sum_(r=0)^(n-1)((r)/(n^(2) + r^(2)))

Find the sum sum_(i=0)^(r)*^(n_(1))C_(r-i).^(n_(2))C_(i)

The value of lim_( n to oo) sum_(r =1)^(n) (r )/(n^(2))"sec"^(2)(r^(2))/(n^(2)) is equal to -