My 99 day streak problem

Calculus Level 3

\[ \Large \lim_{x\to0} \sum_{r=1}^{10} \frac{\exp[\left(1+ \frac12 + \frac14 + \frac18 + \ldots \right) - 2]}{x \cdot \csc((2r-1)x)} \]

Let \(m\) denote the value of the limit above. Evaluate the expression below.

\[ \Large \lim_{x\to0} \ln \left [ \left( e^{(m-1)x}\right)^{\frac{ \sin(x)}{x^2} } \right ] \]

Note: \( \exp A = e^A \).

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