# Continued Fraction for $e$

Does anyone know of some nice/simple proofs for the continued fraction of $e$?

i.e. $$e = [2;1,2,1,1,4,1,1,6,...,1,1,2k,1,1,...]$$

I have read a nice method in

Cohn, H. "A Short Proof of the Simple Continued Fraction Expansion of e." Amer. Math. Monthly 113, 57-62, 2006.

but I am not satisfied with the (for the time being) justification for introducing three seemingly random integrals as this is not within my current scope of understanding!

I've some time ago been interested in the same thing and after finding an article of Gosper (see "Hakmem") I noted the following, which might also be interesting: perhaps a path to a proof for your "special case" $e^1$ can be derived.

> (...)
> http://www.inwap.com/pdp10/hbaker/hakmem/cf.html#item101b
>
>Keith Ramsay

The Gosper-article inspired me to check some powers of e.
If you allow negative coefficients, you seem to get a pretty general scheme
for

1/k
e      for k= ... -3,-2,-1,{0},1,2,3,...

------------------------------------------------------------------------------------------
-       1            3            5            7            9           11
------------------------------------------------------------------------------------------
cf(e^(1/-2)):    [1,-3, 1,    1, -7, 1,    1,-11, 1,    1,-15, 1,    1,-19, 1,    1,-23, 1 ]
cf(e^(1/-1)):    [1,-2, 1,    1, -4, 1,    1, -6, 1,    1, -8, 1,    1,-10, 1,    1,-12, 1 ]

cf(e^(1/ 0)):    [1,-1, 1,    1, -1, 1,    1, -1, 1,    1, -1, 1,    1, -1, 1,    1, -1, 1 ]  divergent (oscillates on 0 and 1)

cf(e^(1/ 1)):    [1, 0, 1,    1,  2, 1,    1,  4, 1,    1,  6, 1,    1,  8, 1,    1, 10, 1 ]
cf(e^(1/ 2)):    [1, 1, 1,    1,  5, 1,    1,  9, 1,    1, 13, 1,    1, 17, 1,    1, 21, 1 ]
cf(e^(1/ 3)):    [1, 2, 1,    1,  8, 1,    1, 14, 1,    1, 20, 1,    1, 26, 1,    1, 32, 1 ]
cf(e^(1/ 4)):    [1, 3, 1,    1, 11, 1,    1, 19, 1,    1, 27, 1,    1, 35, 1,    1, 43, 1 ]
cf(e^(1/ 5)):    [1, 4, 1,    1, 14, 1,    1, 24, 1,    1, 34, 1,    1, 44, 1,    1, 54, 1 ]
cf(e^(1/ 6)):    [1, 5, 1,    1, 17, 1,    1, 29, 1,    1, 41, 1,    1, 53, 1,    1, 65, 1 ]
------------------------------------------------------------------------------------------
+       1            3            5            7            9           11
------------------------------------------------------------------------------------------


for
2/k
e      for k= ... -3,-2,-1,{0},1,2,3,...
-----------------------------------------------------------------------------------------------------------------------
delta:      -       1  12   5           7   36  11           13   60  17            19   84   23           25
-----------------------------------------------------------------------------------------------------------------------
cf(e^(2/-1));   [1,-1, -6, -3, 1,   1, -4, -18, -6, 1,    1, -7, -30, -9, 1,    1, -10, -42, -12, 1,    1,-23  ]
cf(e^(2/1));    [1, 0,  6,  2, 1,   1,  3,  18,  5, 1,    1,  6,  30,  8, 1,    1,   9,  42,  11, 1,    1, 12  ]
cf(e^(2/3));    [1, 1, 18,  7, 1,   1, 10,  54, 16, 1,    1, 19,  90, 25, 1,    1,  28, 126,  34, 1,    1, 37  ]
cf(e^(2/5));    [1, 2, 30, 12, 1,   1, 17,  90, 27, 1,    1, 32, 150, 42, 1,    1,  47, 210,  57, 1,    1, 62  ]
cf(e^(2/7));    [1, 3, 42, 17, 1,   1, 24, 126, 38, 1,    1, 45, 210, 59, 1,    1,  66, 294,  80, 1,    1, 87  ]
cf(e^(2/9));    [1, 4, 54, 22, 1,   1, 31, 162, 49, 1,    1, 58, 270, 76, 1,    1,  85, 378, 103, 1,    1, 112 ]
cf(e^(2/11));   [1, 5, 66, 27, 1,   1, 38, 198, 60, 1,    1, 71, 330, 93, 1,    1, 104, 462, 126, 1,    1, 95  ]
-----------------------------------------------------------------------------------------------------------------------
delta:      +       1  12   5           7   36  11           13   60  17            19   84   23           25
-----------------------------------------------------------------------------------------------------------------------


Also, allowing fractions for coefficients, the primary expansion of e = e^(1/1) = e^(2/2) can be inserted in the previous table:

-----------------------------------------------------------------------------------------------------------------------
...
cf(e^(2/1)):    [1, 0,    6,  2,   1,   1,  3,   18,  5,   1,    1,  6,   30,  8,   1,    1,  9,   42,  11,   1 ...
cf(e^(2/2)):    [1, 0.5, 12,  4.5, 1,   1,  6.5, 36, 10.5, 1,    1, 12.5, 60, 16.5, 1,    1, 18.5, 84,  22.5, 1 ...
cf(e^(2/3)):    [1, 1,   18,  7,   1,   1, 10,   54, 16,   1,    1, 19,   90, 25,   1,    1, 28,  126,  34,   1 ...
...
-----------------------------------------------------------------------------------------------------------------------
delta        +      0.5   6   2.5           3.5  18   5.5            6.5  60   8.5            9.5  42   11.5
-----------------------------------------------------------------------------------------------------------------------

Perhaps this allowing of negative and/or fractional coefficients enables also to find
more simple regularities for e^k  with abs(k)>2

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