Series and CFs

$C_n$ is the $n$ th 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

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

adamponting.com