Continued Fraction Expansion of Euler's Number/Convergents

Theorem
The convergents of the continued fraction expansion to Euler's number $e$ are:
 * $2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}, \dfrac {1264} {465}, \dfrac {1457} {536}, \dfrac {2721} {1001}, \ldots$

These best rational approximations are accurate to $0, 0, 1, 1, 2, 3, 3, 4, 5, 5, \ldots$ decimals.

The fraction $\dfrac {878} {323}$ is exceptionally easy to remember:
 * $\dfrac {878} {323} = 2 \cdotp 71826 \, 625 \ldots$

although this does not occur in the above continued fraction expansion.