G285 2021 Fall Problem Set Problem 8
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Problem
Find
\[\textbf{(A)}\ \frac{16}{27} \qquad \textbf{(B)}\ \frac{32}{27} \qquad \textbf{(C)}\ \frac{64}{27} \qquad \textbf{(D)}\ \frac{128}{27} \qquad \textbf{(E)}\ \frac{256}{27} ==Solution== We begin with a simpler problem\] (Error compiling LaTeX. Unknown error_msg)
\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{1}{4^{(a+b+c)}}$$ (Error compiling LaTeX. Unknown error_msg). Now, suppose and are constant. We have a converging geometric series for with a sum of . Now, make everchanging. We have , so the entire sum must be .
Now, coming back to the original problem, we split the single sum into : Split into single variables to get Now, generalize to obtain . Using the geometric series formula we have Now, we can plug this in for all to get