Product of sums

Calculus Level 4

$\displaystyle \large f(x) = \prod _{k=0}^{\infty}\left(\sum _{n=0}^{A-1}x^{nA^k}\right)$

Find the value of $$f \left(\frac 12 \right)$$, given that $$A$$ is an integer greater than 1.

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