Infinite Series and Taylor Methods

📋 Course Outline

1. Sequence Limits and Convergence
2. Geometric and Infinite Series
3. Integral and P-Series Tests
4. Direct and Limit Comparison
5. Alternating and Absolute Convergence
6. Choosing a Convergence Test
7. Power Series Representations
8. Taylor and Maclaurin Series
9. Standard Maclaurin Series
10. Taylor Approximation and Error

📖 1. Sequence Limits and Convergence

🔑 Key Concepts & Definitions

• Infinite sequence : An ordered list of numbers, written as a_n, and can be viewed as a function whose domain is the positive integers or another specified integer index set.
• Sequence convergence : A sequence converges to L when its terms become arbitrarily close to L as n approaches infinity; if no finite limit exists, the sequence diverges.
• Monotonic Sequence Theorem : Every bounded monotonic sequence converges; in particular, an increasing sequence bounded above or a decreasing sequence bounded below converges.

📝 Essential Points

• The sequence r^n converges precisely when -1 < r <= 1, with limit 0 for -1 < r < 1 and limit 1 for r = 1.

📖 2. Geometric and Infinite Series