# Continued Fraction Expansion of Euler's Number

## Theorem

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

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

### Convergents

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

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

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

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.