📘 Calculus 2: Infinite Series Study Notes

1. Sequences vs. Series

• Sequence ${a_n}$ : An ordered list of numbers. Convergence means $\lim_{n \to \infty} a_n = L$ .

• Series $\sum a_n$ : The sum of the terms.

2. Geometric Series

• General Form: $\sum_{n=0}^{\infty} ar^n$

• Convergence: If |r| < 1 .

• Sum Formula:$S = \frac{a}{1 - r}$

3. P-Series Test

• General Form: $\sum_{n=1}^{\infty} \frac{1}{n^p}$

• Convergence: If p > 1.

• Harmonic Series: $\sum \frac{1}{n} (p = 1)$ , which always diverges.

4. nth Term Test (Divergence Test)

• The Rule: If $\lim_{n \to \infty} a_n \neq 0$ , then the series \sum $a_n$ diverges.

• Important: If the limit is 0, the test is inconclusive.

5. The Integral Test

• The Rule: $\sum a_n and \int_{1}^{\infty} f(x) dx$ behave the same way.

• Requirements: $f(x)$ must be Positive, Continuous, and Decreasing for $x \geq 1$ .

6. Comparison Tests

• Direct Comparison (DCT): $* If a_n \leq b_n$ and $\sum b_n$ converges, then$\sum a_n$ converges.

  • If $a_n \geq b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

• Telescoping Series: Use Partial Fraction Decomposition to find a pattern where middle terms cancel, leaving a finite sum $S_n.$

🗂 Flashcards (Clean LaTeX for Knowt)

Front: Geometric Series Sum formula

Back:$\frac{a}{1 - r}$

Front: P-Series Convergence condition

Back: p > 1

Front: Harmonic Series formula

Back: $\sum_{n=1}^{\infty} \frac{1}{n}$

Front: Direct Comparison (proving convergence)

Back:$0 \leq a_n \leq b_n and \sum b_n$ converges .