(vi)(2^(n)+2^(n-1))/(2^(n+1)-2^(n))=(3)/(2)

If A=([x,x],[x,x]) then A^(n)(n in N)= 1) ([2^nx^n,2^nx^n],[2^nx^n,2^nx^n]) 2) ([2^(n-1) x^n,2^(n-1) x^n],[2^(n-1) x^n,2^(n-1) x^n]) 3) I 4) ([2^(n) x^(n-1),2^(n) x^(n-1)],[2^(n) x^(n-1),2^(n) x^(n-1)])

lim_(n rarr oo)(3^(n+1)+2^(n+2))/(3^(n-1)+2^(n-2)) =

1^(2)C_(1)-2^(2)C_(2)+3^(2)C_(3)-+(-1)^(n-1)n^(2)C_(n)=(1)(n^(2)*2^(n+1))/(n+1)(3)(2^(n+1))/(n-1)

A) |lim_(n rarr oo)((n^((1)/(2)))/(n^((3)/(2)))+(n^((1)/(2)))/((n+3)^((3)/(2)))+....+(n^((1)/(2)))/( n+3(n-1) ^((3)/(2))))=

lim_(n rarr oo) [(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+(n+3)/(n^(2)+3^(2))+.....+(1)/(n)]

lim_(n rarr oo) [(n+1)/(n^(2)+1^(2) )+(n+2)/(n^(2)+2^(2))+(n^+3)/(n^(2)+3^(2))+.....1/n] =

underset(nrarroo)lim[(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+(n+3)/(n^(2)+3^(2))+...+(1)/(n)]

lim_(nrarroo)[(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+(n+3)/(n^(2)+3^(2))+...+(1)/(n)]

Find the sum of series upto n terms ((2n+1)/(2n-1))+3((2n+1)/(2n-1))^(2)+5((2n+1)/(2n-1))^(3)+