Pre-cursor to natural base \(e\)

For the special case when \(n = 1\) and \(R = 100\), it is interesting to examine what happens to \(A\) as \(k\) tends to infinity.

That is, what is the limiting value of
\[ A = P \left( 1+\frac{1}{k} \right)^{k} \]
as \(k \to \infty\)?

It turns out that
\[ \lim_{k \to \infty} \left( 1 + \frac{1}{k} \right)^k \]
exists as it is monotonically increasing and bounded above. We denote the above limit by the constant \(e\), and
\[ e \approx 2.7182818284\ldots \]
This constant is called the Euler's number.

Hence
\[ \lim_{k \to \infty} A = Pe. \]

We shall see later in a later module Differentiation the role the Euler number \(e\) plays.