Evaluation of a class of continued fractions Is there a closed-form way of writing the continued fraction:
$$
1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}}
$$
EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a similar expression for:
$$
2 + \frac{3}{4+ \frac{5}{6 + \frac{7}{8 + ...}}}
$$
More generally, are there general closed-form expressions for all continued fractions of the form:
$$
a_n = n + \frac{n+1}{(n+2) + \frac{n+3}{(n+4) + ...}} \\
f(x) = x + \frac{x+1}{(x+2) + \frac{x+3}{(x+4) + \cdots}} = x + \frac{x+1}{f(x+2)} \\
f(x) f(x+2) = xf(x+2) + x+1
$$
And can said closed form be extended to all real numbers? For example, I experimented with extending the sequence to negative values of n and found that for all negative odd $n, a_n = -1$.

It's $1/(e^{1/2}-1)$.  You should be able to derive this by doing a term-by-term transformation on an appropriate infinite series.
