Calculus Volume 2 · free preview
Alternating Series
An alternating series with terms decreasing to zero in absolute value converges, its error smaller than the first neglected term, and absolute convergence implies convergence while conditional convergence keeps the sign structure essential.
1. The alternating series test
The alternating series test
2. Error estimation
Error estimation
3. Absolute and conditional convergence
Absolute and conditional convergence
4. The alternating series' speed
The alternating series' speed
5. Alternating test hypotheses
The alternating series test for sum of (-1)^(n+1) b_n requires which conditions on b_n?
6. Alternating harmonic
The alternating harmonic series sum of (-1)^(n+1)/n converges, while the plain harmonic diverges. What does this show?
7. Error bound
After N terms of a convergent alternating series, the error is bounded by what quantity?
8. Absolute implies convergence
If the series of absolute values sum |a_n| converges, what follows about the original series sum a_n?
9. Decrease hypothesis necessity
The alternating series test requires b_n to decrease. A series with b_n approaching zero but NOT decreasing can which outcome exhibit?
10. Alternating error at ten terms
For the alternating harmonic series, the error after 10 terms is at most 1/11. As a decimal to three places, what is it? Answer with a number.
11. Conditional example name
A series that converges but not absolutely is said to converge conditionally, and the alternating harmonic series is the model. As one word, what is the stronger kind of convergence? Answer in lowercase.
12. Rearrangement theorem name
The theorem that conditionally convergent series can be rearranged to sum to any value is Riemann's theorem. As a surname, whose theorem? Answer in lowercase.
13. Rearrangement sensitivity
Riemann's theorem says a conditionally convergent series can be rearranged to sum to any prescribed value. Why does this not contradict the alternating series test?
14. Absolute ratio forthcoming
The next section's ratio test decides which kind of convergence mechanically? Answer with one word.
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