# What does Euler's number represent?

Sep 9, 2015

There are many ways to answer that question.

#### Explanation:

It is the limit approached by ${\left(1 + \frac{1}{n}\right)}^{n}$ as $n$ increases without bound.

It is the limit approached by ${\left(1 + n\right)}^{1} / n$ as $n$ approaches $0$ from the right.

It is he number that the sum:

$1 + \frac{1}{1} + \frac{1}{2} + \frac{1}{3 \cdot 2} + \frac{1}{4 \cdot 3 \cdot 2} + \frac{1}{5 \cdot 4 \cdot 3 \cdot 2} + . . .$ approaches as the number of terms increases without bound.

It is the base of the function with y intercept $1$, whose tangent line at $\left(x , f \left(x\right)\right)$ has slope $f \left(x\right)$. This function turns out to be the exponential function $f \left(x\right) = {e}^{x}$.

It is the base for the growth function whose rate of growth at time $t$ is equal to the amount present at time $t$.

It is the value of $a$ for which the area under the graph of $y = \frac{1}{x}$ and above the $x$-axis from $1$ to $x$ equals $1$.
If we define $\ln x$ for $x > + 1$ (as we often do in Calculus 1) as the area from $1$ to $x$ under the graph of $y = \frac{1}{x}$, then $e$ is the number whose $\ln$ is $1$.