Continued Fraction Expansion of Euler's Number

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Theorem

The constant Euler's number $e$ has the continued fraction expansion:

$e = \left[{2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots}\right]$

This sequence is A003417 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Convergents

The convergents of the continued fraction expansion to Euler's number $e$ are:

$2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}, \dfrac {1264} {465}, \dfrac {1457} {536}, \dfrac {2721} {1001}, \ldots$

The numerators form sequence A007676 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The denominators form sequence A007677 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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

This sequence is A114539 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


Proof


Sources