Exercise 4 Page 251

The natural base $e$ is a mathematical constant. It is an irrational number just as $\pi ,$ and therefore we cannot find its exact value. Nevertheless, we can approximate its value in different ways. One of them is the sum shown below. And alternative way involves the expression shown below. As $x\to \infty ,$ the value of the expression shown above approaches $e.$ Let's analyze what happens when $x$ increases. To do this, larger and larger values will be substituted into the expression.

$x$	${\left(1+\frac{1}{n}\right)}^x$	Expression value
$10$	${\left(1+\frac{1}{10}\right)}^{10}$	$2.59374\dots$
$100$	${\left(1+\frac{1}{100}\right)}^{100}$	$2.70481\dots$
$1000$	${\left(1+\frac{1}{1000}\right)}^{1000}$	$2.71692\dots$
$10000$	${\left(1+\frac{1}{10000}\right)}^{10000}$	$2.71814\dots$
$100000$	${\left(1+\frac{1}{100000}\right)}^{100000}$	$2.71826\dots$
$1000000$	${\left(1+\frac{1}{1000000}\right)}^{1000000}$	$2.71828\dots$

The table shows that for high values of $x,$ the expression seems to approach $\sim 2.718.$ The first six digits of this constant are shown below.