the question involves person A eating half a loaf of bread. then person B eats half of the half left over. then person A eats half of whats left over... etc. I defined the series of person A as $\sum_{n=1}^{\infty} \frac{1}{2^{2n-1}}$ and person B as $\sum_{n=1}^{\infty} \frac{1}{2^{2n}}$. How would you sum the 2 series separately so you know how much person A ate and how much person B ate?
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$\begingroup$ Your sums are set up correctly. Hint: These are geometric series. $\endgroup$Brett Frankel– Brett Frankel2014-04-14 01:50:16 +00:00Commented Apr 14, 2014 at 1:50
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2 Answers
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Hint: Just note that $S_B = \frac12 S_A$, and $S_A+ S_B=1$. Can you finish it?
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For person $B$ you have: $$ \sum_{n \ge 1} 2^{-2 n} = 2^{-2} \sum_{n \ge 0} 4^{-n} = \frac{1}{4} \cdot \frac{1}{1 - 1 / 4} = \frac{1}{3} $$ so person $A$ gets: $$ 1 - \frac{1}{3} = \frac{2}{3} $$
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$\begingroup$ You should endeavor to give hints for homework, not just explicitly solve the problem. The student learns nothing from a solution except how to copy. $\endgroup$MPW– MPW2014-04-14 01:28:26 +00:00Commented Apr 14, 2014 at 1:28