Infinite Series and Taylor Methods

📋 Course Outline

1. Sequences and Their Limits
2. Monotone and Bounded Sequences
3. Infinite Series and Geometric Sums
4. Integral and P-Series Tests
5. Comparison Tests and Error Bounds
6. Alternating and Absolute Convergence
7. Choosing a Convergence Test
8. Power Series and Their Calculus
9. Taylor and Maclaurin Series
10. Taylor Approximation and Applications

📖 1. Sequences and Their Limits

🔑 Key Concepts & Definitions

• Sequence : An infinite sequence is an ordered list of numbers whose nth term is commonly written as a_n, and it can be viewed as a function with integer inputs.
• Sequence Limit : A sequence a_n converges to L when its terms become arbitrarily close to L as n becomes sufficiently large; if no finite limit exists, the sequence diverges.

★ Must-know

• For r>0, the sequence 1/n^r approaches 0 as n approaches infinity.

• The sequence r^n converges precisely when -1<r<=1; its limit is 0 for -1<r<1 and 1 for r=1.

Further detail

📌 If a_n approaches L and f is continuous at L, then f(a_n) approaches f(L).

📖 2. Monotone and Bounded Sequences