Continued Fraction Expansion of Pi/Convergents

Theorem
The convergents the continued fraction expansion to $\pi$ (pi) are:
 * $3, \dfrac {22} 7, \dfrac {333} {106}, \dfrac {355} {113}, \dfrac {103993} {33102}, \dfrac {104348} {33215}$

These best rational approximations are accurate to $0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, \ldots$ decimals.

The fraction $\dfrac {355} {113}$ is exceptionally accurate:
 * $\dfrac {355} {113} = 3 \cdotp 14159 \, 292 \ldots$