Alternating series
A series whose terms switch sign. If the terms shrink steadily towards 0, it converges. The error is smaller than the first term you leave out.
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Analogy
Walking two steps forward and one step back, with steps that shrink: you settle at a point.
Example
Terms shrink to 0, signs alternate.So it converges (to ln 2, about 0.693).S_4 = 1 - 0.5 + 0.333 - 0.25 = 0.583.Error less than 1/5 = 0.2.
In practice
Lets you estimate sums with a guaranteed error bound.
Practice
Flashcards
- When does an alternating series converge?When its terms shrink steadily to 0.
Questions people ask
What is the difference between absolute and conditional convergence?
A series converges absolutely if the sum of the absolute values converges. It converges conditionally if the series converges only because of sign cancellation. The alternating harmonic series converges, but 1 + 1/2 + 1/3 and so on does not, so it is conditional.
How accurate is a partial sum of an alternating series?
The error is smaller than the first term you left out, and the true sum sits between consecutive partial sums. With 1 - 1/2 + 1/3 - 1/4, the next term is 1/5, so the answer is within 0.2 of 0.583.