Continued Fraction Expansion of Euler's Number/Proof 1/Lemma

Theorem

 * For $n \in \Z, n \ge 0$:

Proof
To prove the assertion, we begin by demonstrating the relationships hold for the initial conditions at $n = 0$:

The final step needed to validate the assertion, we must demonstrate that the following three recurrence relations hold:

To prove the first relation, we note that the derivative of the integrand of $A_n$ is equal to the sum of the integrand of $A_n$ with the integrand of $B_{n - 1}$ and the integrand of $C_{n - 1}$.

By integrating both sides of the equation, we verify the first recurrence relation:

To prove the second relation, we note that the derivative of the integrand of $C_n$ is equal to the sum of the integrand of $B_n$ with two times n times the integrand of $A_{n}$ minus the integrand of $C_{n - 1}$.

By integrating both sides of the equation, we verify the second recurrence relation:

To prove the third relation, we have:

From the first relation, combined with the initial condition at $n = 0$ being satisfied, we have:

From the second relation, combined with the initial condition at $n = 0$ being satisfied, we have:

From the third relation, combined with the initial condition at $n = 0$ being satisfied, we have: