Each term of the infinite series below is the square of the sum of a power of $\tfrac12$ and the same power of $\tfrac13$. What is the sum of the series $\sum_{n=1}^{\infty} \left(\dfrac{1}{2^n} + \dfrac{1}{3^n}\right)^2 = \left(\dfrac12 + \dfrac13\right)^2 + \left(\dfrac14 + \dfrac19\right)^2 + \left(\dfrac18 + \dfrac{1}{27}\right)^2 + \dots$