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Calculus Level 3

\[\large f(x)=\prod _{ m=0 }^{ \infty }{\frac {\ \ \displaystyle \sum _{ n=0 }^{ m }{ { x }^{ n }\ \ } }{\displaystyle \sum _{ n=0 }^{ \infty }{ { x }^{ n } } } } \]

Given \(f(x)\) defined as above, define \(C=\left\lfloor { 2 }^{ 120 } \times f\left(\frac { 1 }{ 2 } \right) \right\rfloor. \) Then, how many 1's does the binary representation of \(C\) contain? That is, what is the binary digit sum of \(C?\)

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Hint: You do not need a computer to solve this problem, nor do you need to calculate \(f\left(\frac{1}{2}\right)\) to a high accuracy. There is a more "contest-friendly" way to solve this problem.


Bonus: Can you give a formula for the digit sum of \(\left\lfloor { B }^{ 120 } \times f\left(\frac { 1 }{ B } \right) \right\rfloor\) in base \(B?\)

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