Alternancia entre testes de séries

An alternating series is a series $\sum a_{n}$ in which the $a_{n}$ flip signs; ie, consecutive terms have different signs.

It turns out there is a relatively simple test which can show that certain alternating series are convergent.

First, notice that we can write any alternating series as $\sum (-1)^{n} a_{n}$, where the $a_{n} > 0$ for all $n$. Recall that if $\sum (-1)^{n} a_{n}$ converges, then $\lim_{n \rightarrow \infty} (-1)^{n} a_{n} = 0$; or equivalently, $\lim_{n \rightarrow \infty} a_{n} = 0$. It turns out that one additional condition is sufficient to show that an alternating series converges:

Theorem.(Alternating series test) Let $\sum(-1)^{n} a_{n}$ be an alternating series, where all the $a_{n} > 0$. Then this series converges if $\lim_{n \rightarrow \infty} a_{n} = 0$ and $a_{n}$ is a decreasing sequence; ie, $a_{n+1} \leq a_{n}$ for all $n$.

Of course, since convergence/divergence is not affected by a finite number of terms, the alternating series test is still valid if the hypotheses are satisfied for all $n$ past a certain point. In other words, the alternating series test is still applicable if a finite number of terms don't alternate signs, or a finite number of terms don't form a decreasing sequence.

This last example shows that the additional hypothesis that $a_{n}$ be a decreasing sequence is essential. Without it, the alternating series test is not applicable, and you will have to find some other way to determine the convergence or divergence of a series.

(TODO: add stuff about error bound and proof of the alternating series test).

Overview of alternating series test

An alternating series $\sum a_n$ is a series whose terms alternate from positive to negative. The alternating series test says that an alternating series definitely converges if $|a_n| \geq |a_{n+1}|$ for all $n$, and $\lim_{n\to\infty} a_n = 0$. (For example, the alternating harmonic series, $\frac11 - \frac12 + \frac13 - \frac14 \pm \dots$, converges, even though the harmonic series diverges.) In other words, the test applies to alternating series in which the summands $a_n$ are "decreasing in absolute value and converging to $0$ as $n\to\infty$". However, the test is inconclusive otherwise.