Series and CFs

C_n is the nth convergent.

Euler-Minding series

Since for n\ges 1

    \[P_nQ_{n-1}-Q_nP_{n-1}=(-1)^{n+1}a_1a_2\dots a_n\]

Then, if Q_nQ_{n-1}\neq 0,

    \[C_n-C_{n-1}=(-1)^{n+1}\frac{a_1a_2\dots a_n}{Q_nQ_{n-1}}\]

If Q_i\neq 0 for 1\les i \les n, then

    \[C_n=b_0+\sum\limits_{i=1}^n (-1)^{i+1} \frac{a_1a_2\dots a_i}{Q_{i-1}Q_i}\]

This is the n^{th} partial sum of the Euler-Minding series

    \[b_0+\sum\limits_{i=1}^\infty (-1)^{i+1} \frac{a_1a_2\dots a_i}{Q_{i-1}Q_i}\]

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