Consider a series whose n^ th term is given by T_n = 2n + 1 1^3 + 2^3 + + n^3 . If S_N denotes the sum of the first N terms of this series, then the least integer value of N for which S_N > 3.99 is:

Step-by-step solution

The denominator of the n^ th term is the sum of the cubes of the first n natural numbers, which is n^2(n+1)^2 4 . Thus, the n^ th term can be written as: T_n = 2n + 1 n^2(n+1)^2 4 = 4(2n + 1) n^2(n+1)^2 Notice that the numerator 2n + 1 can be expressed as the difference of squares: (n+1)^2 - n^2 . T_n = 4 [ (n+1)^2 - n^2 n^2(n+1)^2 ] = 4 ( 1 n^2 - 1 (n+1)^2 ) This is a telescoping series. The sum of the first N terms is: S_N = _ n=1 ^ N T_n = 4 [ ( 1 1^2 - 1 2^2 ) + ( 1 2^2 - 1 3^2 ) + + ( 1 N^2 - 1 (N+1)^2 ) ] All