Limits up to the Limit -1!

Calculus Level 4

\[ \lim_{n\to\infty} \left[ \tan \left( \frac x2 \right ) \sec(x) + \tan \left( \frac x{2^2} \right) \sec \left( \frac x2 \right) \\ + \tan\left( \frac x{2^3} \right) \sec \left( \frac x {2^2} \right) +\ldots + \tan \left( \frac x {2^n} \right) \sec \left( \frac x {2^{n-1}} \right) \right ]\]

Let \(f(x) \) denote the limit above. Find the value of \(f\left ( \frac \pi 4 \right) \).

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