๐ Topic Summary
Calculus series are fundamental building blocks in understanding more advanced calculus concepts. They represent the sum of an infinite number of terms, following a specific pattern or rule. Understanding convergence and divergence of these series is crucial for approximations, solving differential equations, and various applications in physics and engineering. This worksheet provides a quick review and practice on these essential ideas.
๐ง Part A: Vocabulary
Match each term with its correct definition:
- Term: Series
- Term: Convergence
- Term: Divergence
- Term: Partial Sum
- Term: Remainder
Definitions:
- The difference between the actual sum and a partial sum of a series.
- A sequence formed by adding the terms of a series up to a certain point.
- A sum of infinitely many terms, often following a specific pattern.
- The property of a series where its partial sums approach a finite limit.
- The property of a series where its partial sums do not approach a finite limit; they increase/decrease without bound or oscillate.
Write the correct definition number next to each term:
| Term |
Definition # |
| Series |
|
| Convergence |
|
| Divergence |
|
| Partial Sum |
|
| Remainder |
|
โ๏ธ Part B: Fill in the Blanks
A series $\sum_{n=1}^{\infty} a_n$ ________ if its sequence of partial sums {$S_n$} approaches a ________ limit as $n$ approaches infinity. If the sequence of partial sums does not approach a finite limit, the series ________. Testing for ________ and ________ is a fundamental skill in calculus.
๐ค Part C: Critical Thinking
Explain, in your own words, why understanding the convergence or divergence of an infinite series is important in practical applications like numerical approximations or modeling physical phenomena. Give a specific example.