(1)   If T_k-T _k-1= k(k-1), where T₀=0,

Then find the value of 

lim (n->infinity)  Summation (k=2 to n) (k+1/T_k) 

T_k - T_k-1=k²-k

T_k-1-T_k-2=(k-1)²-(k-1)

.......

T₁ - T₀ = 1²-1

_______________

T_k - T₀=[k(k+1)(2k+1)/6] - [k(k+1)/2]

            =[k(k-1)(k+1)]/3

(k+1)/T_k =3[1/(k-1)] - 3[1/k]

lim_n->inf ∑^{_{k=2 to inf}} (k+1)/T_k = 3[lim_n->inf ∑^{_{k=2 to inf}}[1/(k-1)] - 3[1/k]

                                                      =3