Theorem 1 in Khinchin's "Continued Fractions" I'm reading an English translation of Khinchin's Continued Fractions and I may have found an error in Theorem 1, page 4.
Khinchin observes that if we simplify a finite continued fraction $[a_0; a_1, ... a_n]$ as $p/q$, and the continued fraction $[a_1; a_2, ... a_n]$ as $p'/q'$, then because of $p/q = a_0 + \frac 1 {p'/q'}$, we can choose values of $p$ and $q$ such that:
$$p = a_0 p' + q', \quad q= p' \tag {*}$$
And uses this to recursively define a standard representation of a finite continued fraction in the form $p/q$.
Khinchin defines $[a_0; a_1, ... a_k]$ as the $k$-th convergent of a continued fraction $[a_0; a_1, ... a_n]$ where $n\geq k$.
Theorem 1 states, where $p_k/q_k$ is the standard representation of the $k$-th convergent:
$$p_k = a_k p_{k-1} + p_{k-2}$$
$$q_k = a_k q_{k-1} + q_{k-2}$$
Now Khinchin writes $p'_r/q'_r$ for the $r$-th convergent. This is where I first got suspicious... isn't he already writing $p_r/q_r$ for the that? Then he says:

On the basis of the formulas in $(*)$,
  $$p_n = q_0p'_{n-1} + q'_{n-1}$$
  $$q_n=p'_{n-1}$$

What? That isn't what $(*)$ says at all! That recurrence relates $[a_0; a_1, ... a_n]$ to $[a_1; a_2, ... a_n]$, here Khinchin is trying to relate $[a_0; a_1, ... a_n]$ to $[a_0; a_1, .. a_{n-1}]$. It seems to me that Khinchin or the translator is using the word "convergent" wrongly. The recurrence relations forming the basis of Theorem 1 seem to be true when "convergent" is interpreted using the definition that I gave above, but I only verified that for $k=2$.
 A: The problem turned out to be a very simple reading error on my part.
In fact Khinchin's $p'_r/q'_r$ is not the $r^{\rm th}$ convergent of the original continued fraction 
$$[a_0; a_1, ... a_n],$$
but of the continued fraction:
$$[a_1; a_2, ..., a_n],$$
in which case his claim is obvious.
A: in the middle of page 3, not numbered, he says

$$  [a_0; a_1, \ldots, a_n] = [a_0; r_1] = a_0 + \frac{1}{r_1}.  $$
  Here, $$ r_1 = [a_1; a_2, \ldots, a_n]  $$

which is consistent with your (*), this being formula (6) on page 4.
Meanwhile, I have gotten good use out of these for years, I recommend you get accustomed to this slang for writing the convergents to some $\sqrt N.$ Notice how the first two convergents are the (legal) $0/1$ and the illegal $1/0.$ Here, for convergent $p/q,$ the integer directly below is $p^2 - 13 q^2.$
$$   \sqrt {13}  $$
$$  
\begin{array}{cccccccccccccccccccccccccccccc}
 & & 3 & & 1 & & 1 & & 1 & & 1 & & 6 & & 1 & & 1 & & 1 & & 1 & & 6 & \\
\frac{0}{1} & \frac{1}{0} & & \frac{3}{1} &  &  \frac{4}{1} & &  \frac{7}{2}  & & \frac{11}{3} & &  \frac{18}{5} & &   \frac{119}{33}  & &   \frac{137}{38}  & &   \frac{256}{71}   & &   \frac{393}{109}  & &   \frac{649}{180}   & &   \frac{4287}{1189}  \\
              \\
 & 1 & & -4 &  &  3 & &  -3  & & 4 & &  -1 & &   4  & &   -3  & &   3   & &   -4  & & 1   & &  -4  
\end{array}
$$
