G285 2021 Fall Problem Set Problem 8
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Problem
If the value of can be represented as , where and are relatively prime. 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 $\frac{1}{4^{b+c})=\left(\frac{4}{3} \right)^2 = \frac{16}{9}$ (Error compiling LaTeX. Unknown error_msg), so the entire sum must be .
Now, coming back to the original problem, we split the single sum into :