My textbook claims that the best rational approximations (relative the size of the numerator and denominator) of an irrational number by using continued fraction are those whose expansions are terminated after a relatively large number.
Concretely, we know that $\pi = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, \ldots]$
So my textbook claims that good approximations would be \begin{align*} x_5 & = [3, 7, 15, 1, 292] = \frac{103993}{33102} & |\pi - x_5| = 5.78\cdot 10^{-10}\\ x_{13} & = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14] = \frac{80143857}{25510582} & |\pi - x_{13}| = 5.79\cdot 10^{-16} \end{align*}
But isn't it just the case that \begin{align*} x_4 & = [3, 7, 15, 1] = \frac{355}{113} & |\pi - x_4| = 2.67\cdot 10^{-7}\\ x_{12} & = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1] = \frac{5419351}{1725033} & |\pi - x_{12}| = 2.21\cdot 10^{-14} \end{align*} are also pretty good relative to the size of the numerator and denominator?
The CFEs are the best rational approximation to a number in the sense that if $\frac pq\in\mathbb Q$ is a convergent of $x\in\mathbb R$ and $\gcd(p,q) = 1$ then $$\left|\frac pq - x\right| = \min_{p'\in\mathbb Z, 1\le q'\le q} \left|\frac{p'}{q'} - x\right|$$ So yes, all convergents are good approximations.