Let _ r=1 ^n r^4 (2 r-1)(2 r+1) = n^3 A + n^2 B + 5 n C + f(n) D (A, B, C, D, N) , where f(n) is the ratio of two linear polynomials such that Lim _ n f(n)= 1 2 . Find the value of (A+B+C+D)

Step-by-step solution

T_r= r^4 (2 r-1)(2 r+1) = 1 16 [ 16 r^4-1+1 (2 r-1)(2 r+1) ] T _ r = 1 16 [ (4 r ^2-1 ) (4 r ^2+1 )+1 4 r ^2-1 ]= 1 16 [ (4 r ^2+1 )+ 1 (2 r -1)(2 r +1) ]= 1 16 [ (4 r ^2+1 )+ 1 2 ( (2 r +1)-(2 r -1) (2 r -1)(2 r +1) ) ]= 1 16 [4 r ^2+1+ 1 2 ( 1 (2 r -1) - 1 (2 r +1) ) ] S _ n = 1 4 r ^2+ 1 16 1+ 1 32 ( 1 2 r -1 - 1 2 r +1 )= n ( n +1)(2 n +1) 4 6 + n 16 + 1 32 [ (1- 1 3 )+ ( 1 3 - 1 5 )+ . .+ ( 1 2 n -1 - 1 2 n +1 ) ]= 2 n ^3+3 n ^2+ n 24 + n 16 + 1 32 (1- 1 2 n +1 )= n ^3 12 + n ^2 8 + 5 n 48 + 1 16 ( n 2 n +1 )=