ยง11.2: Series
This section focuses on what a series is, how partial sums $S_n$ determine convergence, and two types of series: telescoping series and geometric series.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| OpenStax Calculus: Infinite Series | Textbook Notes | Best single textbook-style overview here for the definition of an infinite series, the role of partial sums, the meaning of convergence, and introductory geometric and telescoping examples. | Read Notes |
| Paul's Notes: Series — The Basics | Study Guide | Especially useful for the first day of the section: sigma notation, what partial sums are, how a series is defined from the limit of $S_n$, and how to read and rewrite basic series notation. | Read Notes |
| Paul's Notes: Special Series | Focused Notes | A good source for geometric series and telescoping series, with worked examples showing the cancellation pattern in partial sums. | Read Notes |
| Paul's Practice: Series — The Basics | Practice Set | Good for sigma notation, index shifts, writing out terms, and getting comfortable with partial sums before moving on to the named series. | Practice |
| OpenStax Chapter 5 Exercises | Textbook Practice | Mixed exercises from the same chapter, including work with partial sums, convergence from $S_n$, telescoping structure, and geometric series formulas. | Open Section |
| Khan Academy: Series Unit | Practice Hub | A convenient practice hub for partial sums, infinite series, geometric series, and telescoping series, with short videos and quick exercises. | Open Unit |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| Video Calculus Series (Series) | Houston ACT | The UH Video Calculus index explicitly lists a Series video covering sequences of partial sums, geometric series, and the harmonic series. For this section, the partial-sum and geometric parts are the most relevant. | Open Page |
| Partial Sums Intro | Khan Academy | Good first video for understanding $S_n$ notation and the idea that a series is studied through the sequence of its partial sums. | Watch |
| Telescoping Series | Khan Academy | Short worked example showing how cancellation appears after rewriting the terms, which is exactly the main idea students need for basic telescoping problems. | Watch |
| Geometric Series Intro | Khan Academy | Helpful for recognizing the common ratio and seeing where the geometric-series formula comes from before applying it in harder examples. | Watch |
Focused Practice Path
| Skill | Best Resource | What to Focus On | Link |
|---|---|---|---|
| Read series notation and write $S_n$ | Paul's Basics | Make sure you can move back and forth between sigma notation, written-out terms, and the definition of the $n$th partial sum. | Start Here |
| Determine a sum from partial sums | OpenStax | Focus on the core idea: a series converges when the sequence $S_n$ converges, and the value of the series is $\lim_{n\to\infty} S_n$. | Study Method |
| Evaluate telescoping series | Paul's Special Series | Rewrite the general term if needed, write out several terms of $S_n$, and identify exactly which beginning and ending terms survive after cancellation. | Practice Telescoping |
| Recognize and sum geometric series | OpenStax | Train yourself to spot the common ratio quickly, rewrite shifted versions into standard form when needed, and apply the finite/convergent geometric formula correctly. | Practice Geometric |