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

Consider the function $f(n)$ defined by

$f(n) = 1 + (-1)^n.$

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 an $L \in \mathbb{R}$ 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 have $2N > N$ and $2N+1>N$ . Taking $\varepsilon = \frac{1}{2}$ we then have

$\begin{align*} &&|f(2N) - L| &< \frac{1}{2} & \text{and} && | f(2N+1) - L | &< \frac{1}{2} \\[9pt] \implies && |2-L| &< \frac{1}{2} & \text{and} && |-L| &< \frac{1}{2}. \end{align*}$

But then by the triangle inequality this implies

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

a contradiction. Thus, $\{ f(n) \}$ diverges $. \qquad \blacksquare$

Stumbling Robot

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

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