Alternating Series and Absolute Convergence
Handle series whose terms change sign: the alternating test, the error bound from the first omitted term, and the difference between converging absolutely and converging only conditionally.
What a learner can do afterwards
- Apply the alternating series test and bound the truncation error
- Give a series that converges but whose absolute version does not
- Explain why absolute convergence is the stronger claim
1 · Read
An alternating series flips sign every term, and its test asks only two things: magnitudes decrease, and they tend to zero. Pass both and the series converges. The alternating harmonic series is the mascot: 1 over n shrinks to 0, so it converges. Stopping early costs little: the error of any partial sum is at most the first omitted magnitude, so ten terms leave an error of at most 1 over 11.
Strip the signs from the alternating harmonic series and you get the plain harmonic series, which diverges. Converging with signs but diverging without them is called conditional convergence. Contrast minus 1 to the n over n squared: absolute values give 1 over n squared, a p-series with p equals 2, so it converges even naked, which is absolute convergence.
Absolute convergence is the stronger claim: it forces ordinary convergence, but the reverse fails, as the alternating harmonic series shows. So test the absolute version first with your positive-term tools like ratio, and only fall back to the alternating test when those shrug.
Work any signed series in order: confirm the signs truly alternate, confirm magnitudes decrease to zero, then bound your stopping error by the next magnitude.
Alternating plus shrinking to zero converges, the next term bounds the error, and absolute convergence is stronger.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.