Can you find the limit of the product of the sum?

Calculus Level 3

\[ \begin{eqnarray} \displaystyle \sigma(m) &=& \sum_{k=0}^m a^k \\ \displaystyle \rho (n) &=& \prod_{m=1}^n \bigg(1 - \sigma(m) + a \sigma(m) \bigg) \\ \end{eqnarray} \]

Let \(a\) be a real number between 0 and 1 exclusive, and denote the functions as described above.

Find \( \displaystyle \lim_{n\to\infty} \rho(n) \).

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