Continued Fraction Expansion of Euler's Number

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Theorem

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

\(\ds e\) \(=\) \(\ds \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\)
\(\ds \) \(=\) \(\ds \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\)


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 1

From the recursive definition of continued fractions, we have:

\(\ds p_i\) \(=\) \(\ds a_i p_{i - 1} + p_{i - 2}\)
\(\ds q_i\) \(=\) \(\ds a_i q_{i - 1} + q_{i - 2}\)


Let:

\(\ds \sqbrk {a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, \ldots}\) \(=\) \(\ds \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, \ldots}\)

In other words:

$a_{3 n + 1} = 2 n$

and:

$a_{3 n + 0} = a_{3 n + 2} = 1$


Then $p_i$ and $q_i$ are as follows:

$\begin{array}{r|cccccccccc} \ds i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline p_i & 1 & 1 & 2 & 3 & 8 & 11 & 19 & 87 & 106 & 193 \\ q_i & 1 & 0 & 1 & 1 & 3 & 4 & 7 & 32 & 39 & 71 \\ \hline \end{array}$


Furthermore, $p_i$ and $q_i$ satisfy the following $6$ recurrence relations:

\(\text {(1)}: \quad\) \(\ds p_{3 n + 0}\) \(=\) \(\, \ds \paren {a_{3 n + 0} } p_{3 n - 1} + p_{3 n - 2} \, \) \(\, \ds = \, \) \(\ds p_{3 n - 1} + p_{3 n - 2}\)
\(\text {(2)}: \quad\) \(\ds p_{3 n + 1}\) \(=\) \(\, \ds \paren {a_{3 n + 1} } p_{3 n + 0} + p_{3 n - 1} \, \) \(\, \ds = \, \) \(\ds 2 n p_{3 n + 0} + p_{3 n - 1}\)
\(\text {(3)}: \quad\) \(\ds p_{3 n + 2}\) \(=\) \(\, \ds \paren {a_{3 n + 2} } p_{3 n + 1} + p_{3 n + 0} \, \) \(\, \ds = \, \) \(\ds p_{3 n + 1} + p_{3 n + 0}\)
\(\text {(4)}: \quad\) \(\ds q_{3 n + 0}\) \(=\) \(\, \ds \paren {a_{3 n + 0} } q_{3 n - 1} + q_{3 n - 2} \, \) \(\, \ds = \, \) \(\ds q_{3 n - 1} + q_{3 n - 2},\)
\(\text {(5)}: \quad\) \(\ds q_{3 n + 1}\) \(=\) \(\, \ds \paren {a_{3 n + 1} } q_{3 n + 0} + q_{3 n - 1} \, \) \(\, \ds = \, \) \(\ds 2 n q_{3 n + 0} + q_{3 n - 1},\)
\(\text {(6)}: \quad\) \(\ds q_{3 n + 2}\) \(=\) \(\, \ds \paren {a_{3 n + 2} } q_{3 n + 1} + q_{3 n + 0} \, \) \(\, \ds = \, \) \(\ds q_{3 n + 1} + q_{3 n + 0},\)


Our ultimate aim is to prove that:

$\ds \lim_{n \mathop \to \infty} \frac {p_n} {q_n} = e$


In the pursuit of that aim, let us define the integrals:

\(\ds A_n\) \(=\) \(\ds \int_0^1 \frac {x^n \paren {x - 1}^n} {n!} e^x \rd x\)
\(\ds B_n\) \(=\) \(\ds \int_0^1 \frac {x^{n + 1} \paren {x - 1}^n} {n!} e^x \rd x\)
\(\ds C_n\) \(=\) \(\ds \int_0^1 \frac {x^n \paren {x - 1}^{n + 1} } {n!} e^x \rd x\)


Lemma

For $n \in \Z , n \ge 0$:
\(\ds A_n\) \(=\) \(\ds q_{3 n} e - p_{3 n}\)
\(\ds B_n\) \(=\) \(\ds p_{3 n + 1} - q_{3 n + 1} e\)
\(\ds C_n\) \(=\) \(\ds p_{3 n + 2} - q_{3 n + 2} e\)

$\Box$


We assert that $A_n$, $B_n$ and $C_n$ all converge to $0$ as $n \to \infty$:

\(\ds \lim_{n \mathop \to \infty} A_n\) \(=\) \(\ds \frac {\frac {x^{n + 1} \paren {x - 1}^{n + 1} } {\paren {n + 1}!} e^x} {\frac {x^n \paren {x - 1}^n } {n!} e^x}\) Radius of Convergence from Limit of Sequence: Real Case
\(\ds \) \(=\) \(\ds \frac {x \paren {x - 1} } {\paren {n + 1} }\)
\(\ds \) \(=\) \(\ds 0\)
\(\ds \lim_{n \mathop \to \infty} B_n\) \(=\) \(\ds \frac {\frac {x^{n + 2} \paren {x - 1}^{n + 1} } {\paren {n + 1}!} e^x} {\frac {x^{n + 1} \paren {x - 1}^n} {n!} e^x}\) Radius of Convergence from Limit of Sequence/Real Case
\(\ds \) \(=\) \(\ds \frac {x \paren {x - 1} } {\paren {n + 1} }\)
\(\ds \) \(=\) \(\ds 0\)
\(\ds \lim_{n \mathop \to \infty} C_n\) \(=\) \(\ds \lim_{n \mathop \to \infty} B_n - \lim_{n \mathop \to \infty} A_n\)
\(\ds \) \(=\) \(\ds 0\)


We now have:

\(\ds \lim_{n \mathop \to \infty} A_n\) \(=\) \(\, \ds \lim_{n \mathop \to \infty} \paren {q_{3 n} e - p_{3 n} } \, \) \(\, \ds = \, \) \(\ds 0\)
\(\ds \lim_{n \mathop \to \infty} B_n\) \(=\) \(\, \ds \lim_{n \mathop \to \infty} \paren {p_{3 n + 1} - q_{3 n + 1} e} \, \) \(\, \ds = \, \) \(\ds 0\)
\(\ds \lim_{n \mathop \to \infty} C_n\) \(=\) \(\, \ds \lim_{n \mathop \to \infty} \paren {p_{3 n + 2} - q_{3 n + 2} e} \, \) \(\, \ds = \, \) \(\ds 0\)

from which we conclude:

\(\ds \lim_{n \mathop \to \infty} \paren {p_n - q_n e}\) \(=\) \(\ds 0\)
\(\ds \lim_{n \mathop \to \infty} p_n\) \(=\) \(\ds q_n e\)
\(\ds \lim_{n \mathop \to \infty} \frac {p_n} {q_n}\) \(=\) \(\ds e\)

$\blacksquare$


Historical Note

Leonhard Euler analyzed the Ricatti equation to prove that the number e has the continued fraction shown here.

Later, while proving the transcendence of e, Charles Hermite also proved this continued fraction.



Sources