Equivalence of Definitions of Euler's Number

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Theorem

The following definitions of the concept of Euler's Number are equivalent:

Limit of Series

The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ converges to a limit.

This limit is Euler's number $e$.


Limit of Sequence

The sequence $\left \langle {x_n} \right \rangle$ defined as $x_n = \left({1 + \dfrac 1 n}\right)^n$ converges to a limit as $n$ increases without bound.

That limit is called Euler's Number and is denoted $e$.


Base of Logarithm

The number $e$ can be defined as the number satisfied by:

$\ln e = 1$.

where $\ln e$ denotes the natural logarithm of $e$.

That $e$ is unique follows from Logarithm is Strictly Increasing.


Exponential Function

The number $e$ can be defined as the number satisfied by:

$e := \exp 1 = e^1$

where $\exp 1$ denotes the exponential function of $1$.



Proof 1

See Exponential as Limit of Sequence for how $\displaystyle \lim_{n \mathop \to \infty} \left({1 + \frac 1 n}\right)^n = e$ follows from the definition of $e$ as the number satisfied by $\ln e = 1$.

See Euler's Number: Limit of Sequence implies Limit of Series for how $\displaystyle e = \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ follows from $\displaystyle \lim_{n \mathop \to \infty} \left({1 + \frac 1 n}\right)^n = e$.

Now suppose $e$ is defined as $\displaystyle e = \sum_{n \mathop = 0}^\infty \frac 1 {n!}$.

Let us consider the series $\displaystyle f \left({x}\right) = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$.

From Series of Power over Factorial Converges, this is convergent for all $x$.

We differentiate $f \left({x}\right)$ WRT $x$ term by term (justified by Power Series Differentiable on Interval of Convergence), and get:

\(\displaystyle D_x \left({f \left({x}\right)}\right)\) \(=\) \(\displaystyle D_x \left({1}\right) + D_x \left({\frac {x} {1!} }\right) + D_x \left({\frac {x^2} {2!} }\right) + D_x \left({\frac {x^3} {3!} }\right) + \cdots + D_x \left({\frac {x^n} {n!} }\right) + D_x \left({\frac {x^{n+1} } {\left({n+1}\right)!} }\right) + \cdots\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 + 1 + \frac {2 x} {2!} + \frac {3 x^2} {3!} + \cdots + \frac {n x^{n-1} } {n!} + \frac {\left({n+1}\right) x^n} {\left({n+1}\right)!} + \cdots\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n-1} } {\left({n-1}\right)!} + \frac {x^n} {n!} \cdots\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle f \left({x}\right)\) $\quad$ $\quad$

Thus we have:

$D_x \left({f \left({x}\right)}\right) = f \left({x}\right)$

From Derivative of Exponential Function:

$f \left({x}\right) = e^x$

From Derivative of Inverse Function:

$D_x \left({f^{-1} \left({x}\right)}\right) = \dfrac 1 {f^{-1} \left({x}\right)}$

Hence from Derivative of Natural Logarithm Function:

$f^{-1} \left({x}\right) = \ln x$

It follows that $e$ can be defined as that number such that $\ln e = 1$.

Hence all the definitions of $e$ as given here are equivalent.

$\blacksquare$


Proof 2

1 implies 2

See Euler's Number: Limit of Sequence Equals Limit of Series.

$\Box$

2 implies 3

See Euler's Number: Limit of Sequence implies Base of Logarithm.

$\Box$

3 implies 4

Let $e$ be the unique solution to the equation $\ln \left({ x }\right) = 1$.

We want to show that $\exp \left({ 1 }\right) = e$, where $\exp$ is the exponential function.

\(\displaystyle \exp 1 = 0\) \(\iff\) \(\displaystyle \ln \left({ \exp 1 }\right) = \ln \left({ e }\right)\) $\quad$ Logarithm is Injective $\quad$
\(\displaystyle \) \(\iff\) \(\displaystyle 1 = \ln \left({ e }\right)\) $\quad$ Exponential is Inverse of Logarithm and Inverse of Inverse $\quad$

where the final equation holds by hypothesis.

Hence the result.

$\Box$

4 implies 1

Let $e = \exp 1$, where $\exp$ denotes the exponential function.

We want to show that:

$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!} = e$

By definition of $\exp$:

$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!} = \exp 1$

And $\exp 1 = e$ by hypothesis.

Hence the result.

$\blacksquare$


Also see