Sequences

Reading

Section 11.1

Problems

• Practice Problems: 11.1 3, 5, 11, 17, 23, 25, 35, 41
• Practice Problems: 11.1 61, 65, 67, 73, 74, 83
• Problems to turn in: 11.1 16, 26, 48, 70
• Challenge (optional): 11.1 85, 86, 87

Topics to know

1. Definition of a sequence
2. Basic examples of sequences and their visualizations
   • \(a_n = 1-\frac{1}{n}\), \(a_n = (-1)^n n\)
   • Visualize as a graph as well as as points in the plane
3. Sequences defined recursively
   • Newton’s method sequence for \(\sqrt{2}\)
   • \(a_{n+1} = \sqrt{a_n}\), \(a_1 = 2\)
4. Definition of limit of sequence, both intuitive and precise
   • Find the limit of \(\frac{n+1}{n+2}\)
5. Sequences derived from functions (theorem 1)
   • \(\lim_{n\to\infty} n^{1/n}\)
6. Geometric sequence for non-negative \(r\) (example 6)
7. Limit laws for sequences, squeeze theorem
8. A sequence that absolutely converges to 0 also converges to 0
9. Geometric sequence for negative \(r\) (example 8)
10. Using squeeze theorem for convergence of more complicated forms (example 9)
    • \(\lim_{n\to\infty} \frac{R^n}{n!}\)
11. Passing a sequence through a function
    • Use for \(n^{1/n}\)
12. Definition of bounded sequence
13. Convergent sequences are bounded (theorem 5)
14. Bounded monotonic sequences converge (theorem 6)
15. How to use theorem 6 for recursively defined functions:
    • First guess what the limit would be if it existed
    • Show that this limit acts as a bound (proof by induction)
    • Show sequence is increasing/decreasing (proof by induction)
    • Apply for \(a_{n+1} = \sqrt{2a_n}\)