Why This Topic?
Up to now, every convergence test in your toolkit — comparison, ratio, root, integral — was designed for series with nonnegative terms. But many important series have terms that change sign. The alternating harmonic series $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=1-\tfrac12+\tfrac13-\tfrac14+\cdots$ is the simplest example: its terms are not all positive, so none of the nonnegative-term tests apply directly.
Two clean questions arise at the door of any mixed-sign series $\sum a_n$:
- Does the absolute series $\sum |a_n|$ converge?
- Does the original series $\sum a_n$ converge?
Four combinations seem possible a priori. One of them — "$\sum|a_n|$ converges but $\sum a_n$ diverges" — is excluded by the central theorem of this lesson. The remaining three define the trichotomy that classifies every series you will ever meet.
1. Alternating Series and the Leibniz Test
Definition — Alternating Series
A series is alternating if its terms strictly alternate in sign. The standard forms are:
$$\sum_{n=1}^{\infty}(-1)^{n+1}b_n = b_1 - b_2 + b_3 - b_4 + \cdots \qquad (b_n > 0)$$
$$\sum_{n=1}^{\infty}(-1)^{n}b_n = -b_1 + b_2 - b_3 + b_4 - \cdots \qquad (b_n > 0)$$
Suppose $\{b_n\}$ is a sequence of positive numbers satisfying:
- $b_{n+1}\le b_n$ for all $n$ (monotone decrease), and
- $\displaystyle\lim_{n\to\infty}b_n = 0$.
Then the alternating series $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}b_n$ converges.