What is a good approximation for $$\omega=\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$$ This will be used to find $$T=\frac{t}{1-\omega}$$ Without using Lambert's continued fraction
2026-03-26 09:58:03.1774519083
Approximating $\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$
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From Lambert's continued fraction we have that: $$\tan z = \frac{z}{1-\frac{z^2}{3-\frac{z^2}{5-\frac{z^2}{7-\frac{z^2}{\ldots}}}}}$$ holds for any $z$ in a neighboorhood of zero, hence, for small $t$'s: $$\omega = 1-\frac{t}{\tan t}.$$