Divergent Real Sequence to Infinity/Examples/(-1)^n times n

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Example of Divergent Real Sequence to Infinity

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$a_n = \paren {-1} n$


Then $\sequence {a_n}$ is divergent to $\infty$.


However, $\sequence {a_n}$ is neither divergent to $+\infty$ nor divergent to $-\infty$.


Proof

We have that:

$\size {a_n} = n$

Let $H \in \R_{>0}$ be given.

Then by the Archimedean Principle:

$\exists N \in \N: N > H$

and so for $n \ge N$:

$n > H$

Thus $\sequence {a_n}$ is divergent to $\infty$.

But $\sequence {a_n}$ cannot be divergent to $+\infty$ because all its odd terms are negative.

Neither can $\sequence {a_n}$ cannot be divergent to $-\infty$ because all its even terms are positive.

Hence the result.

$\blacksquare$


Sources