Fibonacci Decimals

Algebra Level 4

\[\displaystyle \frac{1}{10^0} + \frac{1}{10^1} + \frac{2}{10^2} + \frac{3}{10^3} + \frac{5}{10^4} + \frac{8}{10^5} + \frac{13}{10^6} + \cdots\]

If the value of the sum above can be expressed as \(\dfrac mn\) for positive coprime integers \(m\) and \(n\), find \(\dfrac{n + 1}{\sqrt{m}}\).

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