Is the sequence $\{(-1)^n 1/n\}$ divergent? Is the sequence $\{(-1)^n{1\over n}\}$ divergent or convergent?

It's convergent to $0$ because for every $\varepsilon$, exists $1/N &lt; \varepsilon$ by archimedean property, so $|1/n|&lt; \varepsilon$ for all $n&gt;N$?
Hint: $-1\le(-1)^n\le1$. Either that, or split it up into two subsequences, depending on the parity of n, and prove that both converge to the same limit.
The sequence converges by the alternating series test.
$$ (-1)^n \over n $$
See a tutorial about alternating series test
http://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx

This fits the criteria for convergence by the alternating series test.