Śleszyński–Pringsheim theorem

Remarks

Proof idea: with the fundamental recurrence formulas ($b_0 = 0$), the denominators satisfy $\abs{B_n} \ge (\abs{a_n} + 1)\abs{B_{n-1}} - \abs{a_n}\abs{B_{n-2}}$. By induction $\abs{B_n} - \abs{B_{n-1}} \ge \abs{a_1 \cdots a_n}$, so $1 = \abs{B_0} \le \abs{B_1} \le \cdots$ and no $B_n$ vanishes. Then $\abs{f_n - f_{n-1}} = \frac{\abs{a_1 \cdots a_n}}{\abs{B_n}\abs{B_{n-1}}} \le \frac{1}{\abs{B_{n-1}}} - \frac{1}{\abs{B_n}}$, and this telescopes: the series $\sum (f_n - f_{n-1})$ converges absolutely with total at most $1$, where $f_0 = 0$. Example: $a_n = 1$ and $b_n = 2$ satisfy the condition, and the value is $\sqrt{2} - 1$. The condition cannot simply be dropped: for $a_n = -1$ and $b_n = 1$ one has $\abs{b_n} < \abs{a_n} + 1$, and the convergents repeat the cycle $-1, \infty, 0$, so the continued fraction diverges.

Going deeper

For a student

This is a basic sufficient condition for convergence, in the same spirit as the comparison test for series. Each map $w \mapsto \frac{a_n}{b_n + w}$ sends the closed unit disc into itself under the hypothesis, which explains the bound $\abs{f} \le 1$. The Van Vleck theorem is another classical criterion, for partial denominators in a sector.