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

by RoRi

February 27, 2016

Consider the function $f(n)$ defined by

$f(n) = \frac{1 + (-1)^n}{n}.$

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

To show $\{ f(n) \}$ is convergent we will use the squeeze theorem to determine its limit. First we note

$0 \leq \frac{1+(-1)^n}{n} \leq \frac{2}{n}$

for all $n$ (since $1+(-1)^n = 0$ if $n$ is odd and $1+(-1)^n = 2$ if $n$ is even). Since

$\lim_{n \to \infty} 0 = \lim_{n \to \infty} \frac{2}{n} = 0,$

by the squeeze theorem we have

$\lim_{n \to \infty} \frac{1+(-1)^n}{n} = 0.$

Stumbling Robot