Calculus Volume 2 · free preview
Comparison Tests
The direct comparison and limit comparison tests benchmark a series against a p-series or geometric series of known behavior, the dominant-terms asymptotic deciding convergence for positive rational-exponential terms.
1. Direct comparison
Direct comparison
2. Limit comparison
Limit comparison
3. Choosing benchmarks
Choosing benchmarks
4. The comparison workflow
The comparison workflow
5. Comparison direction convergence
Direct comparison certifies convergence of sum a_n from 0 at most a_n at most b_n when sum b_n does what?
6. Limit comparison hypothesis
Limit comparison applies when the limit of a_n over b_n equals L with what restriction?
7. Dominant term benchmark
The series sum of (n^2 + 3n)/(2n^4 - n) behaves like which benchmark?
8. Log factor absorption
The series sum of ln n/n^2 converges by comparison with which p-series?
9. Divergence by comparison
The series sum of 1/(n - 0.5) diverges by comparison with which benchmark?
10. Benchmark for one over n squared plus one
The series sum of 1/(n^2 + 1) is benchmarked against which p-series? Answer like sum 1/n^2.
11. Ratio limit for shifted p
Limit comparison of sum 1/(n - 1) with the harmonic sum 1/n gives a ratio limit of what number? Answer with a number.
12. Exponential benchmark
A series with terms n^3/2^n converges, dominated ultimately by which geometric family? Answer like sum 1/2^n raised to a power, described in words.
13. Limit comparison L value
Limit comparison of sum of 1/(n^2 - 5) with sum of 1/n^2 gives a ratio limit L equal to what number?
14. Comparison test hazard
Direct comparison requires the inequality to point the correct way, and choosing the wrong benchmark direction leaves the test what? Answer with one word.
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