In the alternating series test, what is the condition for convergence?

Question

When testing for convergence of a series $a_n$ using the Alternating series test, we need to satisfy the condition that $$|a_{n+1}| \leq |a_n|$$ and that $$\lim\limits_{n\rightarrow\infty} a_n = 0$$.

But I've seen other sources that state that the second condition (the limit) is another series that is just positive terms and leaves out the alternating term (eg, $(-1)^n$). That is, that we need to test $$\lim\limits_{n\rightarrow\infty} b_n = 0$$, where, for example, $$a_n = (-1)^nb_n $$

Which one is it?

Answer 1

It doesn't matter if you use the terms with the (alternating) sign, or just their absolute values since: $$|u_n| \to 0 \iff u_n \to 0$$

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• if $u_n \to 0$, then clearly $|u_n| \to 0$;
• if $|u_n| \to 0$, note that $-|u_n| \le u_n \le |u_n|$.

Answer 2

If we call the sequence in question $a_n = (-1)^nb_n$, then you need to verify that $b_1\ge b_2 \ge b_3\ge\dots \ge 0$ and $b_n\to 0$.

The first condition is that $b_n$ is monotonically decreasing. The second says that $b_n$ converges to $0$.