Financial and graphical meaning

The sequence $(1+1/n)^n$ is not an abstraction: it arises from the problem of compounding interest.

Observation — Financial meaning

Investing $1$ euro at an annual rate of $100\%$ compounded $n$ times a year yields $(1+1/n)^n$ euros at the end of the year: $n = 1$ gives $2$ euros, $n = 12$ gives $\approx 2{,}613$, $n = 365$ gives $\approx 2{,}7146$, while $n\to\infty$ (continuous compounding) gives $e\approx 2{,}71828$.

The graph shows the first terms of the sequence $e_n = (1+1/n)^n$ rising slowly towards the horizontal line $y = e$: the sequence increases, but always remains below its limiting value.

The red points are the terms $e_n = (1+1/n)^n$; the blue line is the limiting value $y = e\approx 2{,}71828$. The sequence is increasing and approaches $e$ from below.

Links

Topics: Sequences
Concepts: Limit of a sequence · Euler’s number
Skills: Interpreting a graph · Modelling