Calculus Volume 2 · free preview
Ratio and Root Tests
The ratio test examines successive term ratios and the root test the nth roots, each certifying absolute convergence when the limit is below 1 and diverging above 1, the mechanical workhorses for factorials and exponentials.
1. The ratio test
The ratio test
2. The root test
The root test
3. Choosing and combining
Choosing and combining
4. The toolkit assembled
The toolkit assembled
5. Ratio test verdict
The ratio test computes rho as the limit of |a_(n+1)/a_n|. Which rho certifies absolute convergence?
6. Factorial growth verdict
The series sum of 2^n/n! converges by which observation?
7. Root test best case
The root test is most directly suited to series whose terms have which structure?
8. Boundary inconclusiveness
Both ratio and root tests are inconclusive at rho = 1. Which series illustrates this boundary behaving differently?
9. Factorial over exponential
The series sum of n!/100^n diverges by which ratio test computation?
10. Ratio for n over 2^n
For sum of n/2^n, the ratio of successive terms is (n + 1)/(2n), tending to what number? Answer with a fraction like 1/2.
11. Root for third power
For sum of (1/3)^n, the nth root of the nth term equals what number? Answer with a fraction like 1/3.
12. Boundary name
The ratio and root tests both decline at rho = 1, and the tests are said to be what? Answer with one word.
13. Root test on n-th power terms
The series sum of (3n/(2n + 1))^n has nth root tending to what value?
14. Test target
Both the ratio and root tests certify which kind of convergence when successful? Answer with one word.
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