📚 Topic Summary
Power series manipulations are crucial in calculus for approximating functions, solving differential equations, and evaluating complex integrals. By applying operations like term-by-term differentiation, integration, and algebraic manipulation, we can transform a known power series into one representing a related function. Understanding these techniques unlocks powerful problem-solving capabilities in advanced calculus.
🧠 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Radius of Convergence |
A. A series where each term is a constant times a power of (x-a). |
| 2. Power Series |
B. The set of all x for which the power series converges. |
| 3. Interval of Convergence |
C. The value R such that the power series converges for |x-a| < R and diverges for |x-a| > R. |
| 4. Term-by-Term Differentiation |
D. Differentiating each term of the power series individually. |
| 5. Term-by-Term Integration |
E. Integrating each term of the power series individually. |
Answer Key: 1-C, 2-A, 3-B, 4-D, 5-E
✏️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided: differentiation, integration, algebraic, power series, function.
A __________ is an infinite series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$. We can manipulate these series through __________, __________, and __________ manipulations to represent a new __________. These techniques are essential tools in calculus.
Answer: power series, differentiation, integration, algebraic, function
🤔 Part C: Critical Thinking
Explain how term-by-term differentiation and integration of a power series can be used to find a power series representation for a related function. Provide an example.