Contradictions between the Alternating Series Test & Divergence Test? By the Alternating Test: The series $\sum_{n=1}^{\infty}(-1)^{n+1}\cdot b_n$ converges if all three of the following conditions are satisfied:


The $b_n$'s are all positive.
The positive $b_n$'s are (eventually) decreasing: $b_n\ge b_{n+1}$ $\forall n\ge N$.
$b_n \to 0$


But in the Divergence Test: Given $\sum_{n=1}^{\infty} a_n$, iff $\lim_{n \to \infty} a_n \neq 0 \implies$  $\sum_{n=1}^{\infty} a_n$ diverges



Now given an arbitrary alternating series:

$$S = \sum_{n=1}^{\infty}(-1)^{n+1}\cdot b_n$$

If we take the limit of $a_n$ in the series above

$$\lim_{n \to \infty}(-1)^{n+1}\cdot b_n = \underbrace{\left(\lim_{n \to \infty }(-1)^{n+1}\right)}_\text{This limit doesn't exist}\left(\lim_{n \to \infty} b_n\right)$$

Therefore by the Divergence test, $S$ should be a divergent series, regardless of the conditions needed for the alternating test. But by the Alternating Series Test, $S$ is a convergent series provided the three conditions stipulated initially are met.

So how is this seeming contradiction resolved, by the alternating series test?
if $a_n b_n$ converges, then this does not imply that $a_n$ converges. In case $(-1)^n b_n$ the first factor is bounded. In this case, if the second factor converges to $0$, the product also converges to $0$.
$$\lim_{n\rightarrow\infty} 1 = \lim_{n\rightarrow\infty} (-1)^n(-1)^n =\left(\lim_{n\rightarrow\infty}(-1)^n\right)\left(\lim_{n\rightarrow\infty} (-1)^n\right).$$