Calculus Volume 2 · free preview
The Divergence and Integral Tests
The divergence test restates the nth term condition, and the integral test compares a positive series with an improper integral, converging together, with the p-series classification as its sharpest application.
1. The divergence test
The divergence test
2. The integral test
The integral test
3. The p-series
The p-series
4. Remainder estimates
Remainder estimates
5. Divergence test direction
The divergence test certifies divergence when which condition holds?
6. Integral test hypotheses
The integral test applies to a series whose terms come from f(n) when f satisfies which conditions?
7. P-series threshold
The p-series sum of 1/n^p converges for which exponents?
8. Remainder upper bound
After N terms of a convergent p-series with p = 2, the remainder R_N is at most which integral?
9. Terms not needed condition
The series sum of (n + 1)/n diverges by which observation?
10. Integral of one over x squared
The improper integral from 1 to infinity of dx/x^2 equals what number? Answer with a number.
11. Harmonic integral divergence
The integral from 1 to infinity of dx/x equals what function of the upper limit? Answer with a function like ln b.
12. Remainder of sum one over n cubed
After N terms of sum 1/n^3, the remainder is at most 1 over (2 N^2). As a power, what is the exponent of N? Answer with a number.
13. Zeta at two
The sum of 1/n^2 equals pi^2/6, a famous value. Which test certifies its convergence without giving the sum?
14. Test sharing
The integral test's conclusion is that the series and integral share which fate? Answer with one word, convergence or divergence.
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