Mathematics Question from JEE Main 2023 · 15 April, Shift 1

If the sum of the series $\left(\frac{1}{2}-\frac{1}{3}\right) +\left(\frac{1}{2^2}-\frac{1}{2\cdot3}+\frac{1}{3^2}\right) +\left(\frac{1}{2^3}-\frac{1}{2^2\cdot3}+\frac{1}{2\cdot3^2}-\frac{1}{3^3}\right) +\left(\frac{1}{2^4}-\frac{1}{2^3\cdot3}+\frac{1}{2^2\cdot3^2}-\frac{1}{2\cdot3^3}+\frac{1}{3^4}\right) +\cdots$ is $\dfrac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are coprime, then $\alpha+3\beta$ is equal to