Continued Fraction for $e$

Question

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!

Answer 1

[comment, no answer]

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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