Printable Power Series General Form Practice Problems

📚 Topic Summary

A power series is a series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$, where $c_n$ are constants, $x$ is a variable, and $a$ is a constant called the center of the series. The general form helps us analyze the convergence and behavior of functions. Understanding the general form is crucial for approximating functions and solving differential equations. Think of it as a polynomial with infinite terms!

Finding the general form of a power series often involves manipulating known series like the geometric series $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ or using term-by-term differentiation or integration. It's all about pattern recognition and algebraic manipulation to get it into that standard $\sum_{n=0}^{\infty} c_n(x-a)^n$ format.

🧠 Part A: Vocabulary

Match the term with its correct definition:

1. Terms:
• 1. Power Series
• 2. Radius of Convergence
• 3. Interval of Convergence
• 4. Center of the Series
• 5. General Form
2. Definitions:
• A. The value 'a' in the power series representation.
• B. The interval where the power series converges.
• C. A series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$.
• D. The distance from the center to the nearest point of divergence.
• E. The standard format allowing analysis of convergence and behavior.

Match the correct term with its definition. Write the number and corresponding letter (e.g., 1-C).

✍️ Part B: Fill in the Blanks

Complete the following sentences about power series:

The general form of a power series is given by $\sum_{n=0}^{\infty} c_n(x-a)^n$, where $c_n$ represents the _________, $x$ is the _________, and $a$ is the _________ of the series. The _________ of convergence determines for which values of $x$ the series converges. Manipulating known series such as the _________ series is a common technique for finding power series representations.

🤔 Part C: Critical Thinking

Explain in your own words why representing a function as a power series can be useful in calculus. Provide a specific example of how it simplifies a calculation (e.g., integration, differentiation).