Worked exercise — Limit of (1 - 1/n)^n

Problem

Show that $a_n = \left(1 - \dfrac{1}{n}\right)^n \to \dfrac{1}{e}$.

Solution

We rewrite the exponent, separating out the factor that recalls the definition of $e$. Setting $m = n-1$: $\left(1 - \frac1n\right)^n = \left(\frac{n-1}{n}\right)^n = \left(\frac{n}{n-1}\right)^{-n} = \left(1 + \frac{1}{n-1}\right)^{-n}.$ Now $\left(1 + \frac{1}{n-1}\right)^{-n} = \underbrace{\left(1 + \frac{1}{n-1}\right)^{-(n-1)}}_{\to\, e^{-1}} \cdot \underbrace{\left(1 + \frac{1}{n-1}\right)^{-1}}_{\to\, 1}.$ The first factor tends to $e^{-1}$ (it is the definition of $e$ with exponent and base both $\to n-1$, with the sign changed), the second tends to $1$. Hence $\left(1 - \frac1n\right)^n \to \boxed{\frac{1}{e}}.$

Links

Topics: Sequences
Concepts: Limit of a sequence · Euler’s number
Methods: The number e via a sequence
Skills: Computing limits
Exercise type: Computing a limit