Infinite Sequences and Series

📋 Course Outline

1. Sequence Limits and Behavior
2. Infinite Series and Geometric Sums
3. Integral Test and P-Series
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 Behavior

🔑 Key Concepts & Definitions

• Infinite sequence : An ordered list of numbers, commonly written as $\{a_n\}$, where $a_n$ denotes the nth term and the domain consists of specified positive integers.
• Sequence limit : The statement $a_n\to L$ as $n\to\infty$ means that the terms become arbitrarily close to $L$ for sufficiently large $n$; a finite limit implies convergence.

📝 Essential Points

• The sequence $\{r^n\}$ converges precisely when $-1<r\leq1$, with limit $0$ for $-1<r<1$ and limit $1$ for $r=1$.

• The Monotonic Sequence Theorem states that every bounded monotonic sequence converges, specifically an increasing sequence bounded above or a decreasing sequence bounded below.