I want too calculate the value of this sum:
$$\sum \frac{1}{2^x3^y}$$
Where we sum up all permutations of terms involving a nonnegative integer $x$ and a nonnegative integer $y$.
How can I notationally describe this sum and how can I calculate it (assuming it does converge).
Due to the Fubini/Tonelli Theorem, you can just sum over one index first, and then over the other. That is, $$\sum_{n = 0}^\infty \sum_{m = 0}^\infty \frac{1}{2^n 3^m} = \sum_{n = 0}^\infty \frac 1 {2^n}\left(\sum_{m = 0}^\infty \frac 1{3^y} \right) = \sum_{n = 0}^\infty \frac 1 {2^n} \left(\frac{1}{1-(1/3)}\right) = \frac 3 2 \cdot\frac{1}{1-(1/2)} = 3.$$ It'd be the exact same if you summed over $n$ first, then $m$.