# No computer needed

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 $C?$


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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