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According to Theorem 4.7 in the 2016 Continued Fractions course notes by Gautam Gopal Krishnan, a continued fraction $a_0 + \frac{1}{a_1+}\frac{1}{a_2+}\ldots$ can be rewritten as a series: $$a_0 + \sum_{n=1}^\infty \frac{(-1)^{n-1}}{q_n q_{n-1}}$$ where $q_n$ is given by Theorem 2.4 in the same notes as \begin{eqnarray*} q_0 & = & 1 \\ q_1 & = &...


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