Determine the convergence or divergence of f(n) = cos ( n π / 2)

by RoRi

February 27, 2016

Consider the sequence $f(n)$ defined by

$f(n) = \cos \frac{n \pi}{2}.$

Determine whether the sequence $\{ f_n \}$ converges or diverges, and if it converges find the limit.

The sequence $\{ f(n) \}$ diverges.
Proof. Suppose otherwise, that there exists a number $L \in \mathbb{R}$ and a positive integer $N$ such that

$| f(n) - L | < \varepsilon \qquad \text{for all } \varepsilon > 0 \qquad \text{for all } n > N.$

Since $N$ is a positive integer, we know $4N > N$ and $4N+2 > N$ . But,

$f(4N) = \cos (2N \pi) = 1 \quad \text{and} \quad f(4N+2) = \cos((2N+1)\pi) = -1.$

Taking $\varepsilon = \frac{1}{2}$ we then have

$|1 - L| < \frac{1}{2} \quad \text{and} \quad |-1-L| < \frac{1}{2} \quad \implies \quad |1+L|< \frac{1}{2}.$

But, these imply

$|1-L| + |1+L| < 1.$

By the triangle inequality we then have

$|1 - L + 1 + L | < 1 \quad \implies \quad 2 < 1,$

a contradiction. Hence, there is no such limit $L. \qquad \blacksquare$

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