JEE maths#22

Calculus Level 3

If \(S = \dfrac{4}{1 \cdot 2 \cdot 3} + \dfrac{5}{2 \cdot 3 \cdot 4} + \dfrac{6}{3 \cdot 4 \cdot 5} \cdots \) till \(n\) terms and \(S = K - \dfrac{(2n+5)}{2(n+1)(n+2)} \), then find \( K\).