If the sum of the series \( \left( \dfrac{1}{2} - \dfrac{1}{3} \right) + \left( \dfrac{1}{2^2} - \dfrac{1}{2 \cdot 3} + \dfrac{1}{3^2} \right) + \left( \dfrac{1}{2^3} - \dfrac{1}{2^2 \cdot 3} + \dfrac{1}{2 \cdot 3^2} - \dfrac{1}{3^3} \right) + \left( \dfrac{1}{2^4} - \dfrac{1}{2^3 \cdot 3} + \dfrac{1}{2^2 \cdot 3^2} - \dfrac{1}{2 \cdot 3^3} + \dfrac{1}{3^4} \right) + \ldots \) is \( \dfrac{\alpha}{\beta} \) , where \( \alpha \) and \( \beta \) are co-prime, then \( \alpha + 3\beta \) is equal to _________.
