Difference between revisions of "G285 2021 Fall Problem Set Problem 8"
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==Problem== | ==Problem== | ||
− | Find <cmath>\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{a+2b+3c}{4^{(a+b+c)}}</cmath> | + | Find <cmath>\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \sum_{d=0}^{\infty} \frac{a+2b+3c}{4^{(a+b+c+d)}}</cmath> |
<cmath>\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}</cmath> | <cmath>\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}</cmath> |
Revision as of 21:43, 18 July 2021
Problem
Find
Solution
We begin with a simpler problem . 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