Continued Fraction Expansion of Euler's Number/Convergents

Theorem

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

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

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


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Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$
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