A Sequence So Fine

Calculus Level 3

Given that \(f(x) = 3 \cdot 4^x \cdot 5^{-2x}\), evaluate the expression \[\displaystyle \sum_{a=1}^{\infty} \; f\left (a + \prod_{b=1}^{\infty} f(b) \right ) \; + \; \displaystyle \prod_{c=1}^{\infty} \; f\left (c + \sum_{d=1}^{\infty} f(d) \right ).\]

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