Determine the convergence or divergence of f(n) = (1 + 2/n)ⁿ

by RoRi

February 27, 2016

Consider the function $f(n)$ defined by

$f(n) = \left( 1 + \frac{2}{n} \right)^n.$

Determine whether the sequence $\{ f(n) \}$ converges or diverges, and if it converges find the limit.

By property (10.13) on page 380 of Apostol we know

$\lim_{n \to \infty} \left( 1 + \frac{a}{n} \right)^n = e^a.$

So, in the present case we have

$\lim_{n \to \infty} f(n) = \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^n = e^2.$

Therefore, $\{ f_n \}$ converges to the limit $e^2$ .

Stumbling Robot