Processing math: 50%
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
- Definition of a sequence
- Basic examples of sequences and their visualizations
- an=1−1n, an=(−1)nn
- Visualize as a graph as well as as points in the plane
- Sequences defined recursively
- Newton’s method sequence for √2
- an+1=√an, a1=2
- Definition of limit of sequence, both intuitive and precise
- Sequences derived from functions (theorem 1)
- Geometric sequence for non-negative r (example 6)
- Limit laws for sequences, squeeze theorem
- A sequence that absolutely converges to 0 also converges to 0
- Geometric sequence for negative r (example 8)
- Using squeeze theorem for convergence of more complicated forms (example 9)
- \lim_{n\to\infty} \frac{R^n}{n!}
- Passing a sequence through a function
- Definition of bounded sequence
- Convergent sequences are bounded (theorem 5)
- Bounded monotonic sequences converge (theorem 6)
- 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}