<s>[INST] How to find the solution to a multi-inifite summation? I'm familiar with techniques to solve a single infinite sum such as:
$$
\sum_{i=1}^{\infty} (1 + i) \cdot \frac{1}{2^i}
$$
Which ends up being equal to $3$, but I'm having trouble figuring out how to tackle a multi summation such as this:
$$
\sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i +j) \cdot \left( \frac{1}{3} \right)^{(1+i)} \cdot \left( \frac{2}{3} \right)^{(1+j)}
$$
For the first sum I was able to find a pattern in the partial sums.  But with this double summation, I don't think that will work as easily.  Does anyone know of a technique to solve this second summation analytically? Thanks
 [/INST] A key to calculate this double sum lies in the fact that for absolutely convergent series $\sum_{i=0}^{\infty}a_i$ and $\sum_{j=0}^{\infty}b_j$ we have
$$\left(\sum_{i=0}^{\infty}a_i\right)\left(\sum_{j=0}^{\infty}b_j\right)=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_ib_j = \sum_{j=0}^{\infty}\sum_{i=0}^{\infty}a_ib_j$$
Now, you can just split up the given series and apply above fact
$$\sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i +j) \left( \frac{1}{3} \right)^{(1+i)} \left( \frac{2}{3} \right)^{(1+j)}$$ $$= \underbrace{3\sum_{i=0}^\infty \sum_{j=0}^\infty \left( \frac{1}{3} \right)^{(1+i)} \left( \frac{2}{3} \right)^{(1+j)}}_{=S_1} + \underbrace{\sum_{i=0}^\infty \sum_{j=0}^\infty i  \left( \frac{1}{3} \right)^{(1+i)} \left( \frac{2}{3} \right)^{(1+j)}}_{=S_2} + \underbrace{\sum_{i=0}^\infty \sum_{j=0}^\infty j\left( \frac{1}{3} \right)^{(1+i)} \left( \frac{2}{3} \right)^{(1+j)}}_{=S_3}$$
$$S_1 = 3\cdot \frac 13 \cdot \frac 23 \left(\sum_{i=0}^{\infty}\left( \frac{1}{3} \right)^{i}\right)\left(\sum_{j=0}^{\infty}\left( \frac{2}{3} \right)^{j}\right) = \frac 23\cdot \frac 32 \cdot 3=3$$
$$S_2 =  \frac 23 \left(\sum_{i=0}^{\infty}i\left( \frac{1}{3} \right)^{(1+i)}\right)\left(\sum_{j=0}^{\infty}\left( \frac{2}{3} \right)^{j}\right) \stackrel{\sum_{i=0}^{\infty}i x^{i+1} =\frac{x^2}{(1-x)^2}}{=} \frac 23\cdot \frac 14 \cdot 3 = \frac 12$$
$$S_3 =  \frac 13 \left(\sum_{i=0}^{\infty}\left( \frac{1}{3} \right)^{i}\right)\left(\sum_{j=0}^{\infty}j\left( \frac{2}{3} \right)^{(1+j)}\right) \stackrel{\sum_{j=0}^{\infty}j x^{j+1} =\frac{x^2}{(1-x)^2}}{=} \frac 13\cdot \frac 32 \cdot 4 = 2$$
So, you get 
$$\sum_{i=0}^\infty \sum_{j=0}^\infty (3 + i +j) \left( \frac{1}{3} \right)^{(1+i)} \left( \frac{2}{3} \right)^{(1+j)} = S_1  +S_2 + S_3 = \boxed{\frac{11}{2}}$$
</s>