Difference between revisions of "G285 2021 Fall Problem Set Problem 8"
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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} \frac{a+2b+3c}{4^{(a+b+c)}}</cmath> | ||
| − | < | + | <math>\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}</math> |
==Solution== | ==Solution== | ||
Revision as of 22:37, 11 July 2021
Problem
Find ![\[\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{a+2b+3c}{4^{(a+b+c)}}\]](http://latex.artofproblemsolving.com/e/9/1/e91eecbe57dc0b8011bc0ad39d2b05b34ed54e32.png)
Solution
We begin with a simpler problem ![\[\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{1}{4^{(a+b+c)}}\]](http://latex.artofproblemsolving.com/0/e/0/0e01ec19f235bbece9208e5c66e1ca8dc3963a54.png)
. 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 
: ![\[\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{a}{4^{(a+b+c)}}+\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{2b}{4^{(a+b+c)}}+\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \frac{3c}{4^{(a+b+c)}}\]](http://latex.artofproblemsolving.com/0/d/9/0d9a338ced1a6932c6396bf5e0bfe855bc0227ac.png)
Split into single variables to get ![\[\frac{16}{9} \left(\sum_{a=0}^{\infty} \frac{a}{4^a}+2\sum_{b=0}^{\infty} \frac{b}{4^b} + 3\sum_{c=0}^{\infty} \frac{c}{4^c} \right)\]](http://latex.artofproblemsolving.com/a/d/1/ad181aad6144dadac751e24c8f25a5ea0d95aa0c.png)
Now, generalize 
to obtain 
. Using the geometric series formula we have ![\[\frac{(\tfrac{1}{4}+\tfrac{1}{16}+\tfrac{1}{64}+ \cdots)}{1-\frac{1}{4}} \implies \frac{\tfrac{\tfrac{1}{4}}{1-\frac{1}{4}}}{1-\frac{1}{4}} \implies \frac{4}{9}\]](http://latex.artofproblemsolving.com/7/b/3/7b3d17051ff00a86b51172b2ae6b54e7193fa91f.png)
Now, we can plug this in for all 
to get ![\[\frac{16}{9} \left(6 \cdot \frac{4}{9} \right) \implies \boxed{\textbf{(D)}\ \frac{128}{27}}\]](http://latex.artofproblemsolving.com/e/b/b/ebb0fb2b16f30a936c77c882a0eb5a2d4ae36dcd.png)
