Radius of convergence ratio test worksheets for high school calculus

📚 Topic Summary

The radius of convergence of a power series tells us how far away from the center of the series we can go before the series diverges. The ratio test is a powerful tool to find this radius. It involves taking the limit of the ratio of consecutive terms in the series. If this limit is less than 1, the series converges; if it's greater than 1, the series diverges; and if it's equal to 1, the test is inconclusive. Ultimately, understanding the ratio test helps determine where a power series accurately represents a function.

🧮 Part A: Vocabulary

Match the terms with their correct definitions:

1. Term: Radius of Convergence
2. Term: Power Series
3. Term: Ratio Test
4. Term: Convergence
5. Term: Divergence

Definitions:

1. The behavior of a series that does not approach a finite limit.
2. A test used to determine the convergence of a series by examining the ratio of consecutive terms.
3. An infinite series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$.
4. The distance from the center of a power series to the nearest point where the series diverges.
5. The behavior of a series approaching a finite limit.

(Match 1-5 with A-E, respectively. For example: 1-C)

✍️ Part B: Fill in the Blanks

Complete the following paragraph using the words provided:

(radius, limit, converges, power series, ratio test)

The ________ is a method used to determine the ________ of a ________. This test involves computing the ________ of the absolute value of the ratio of successive terms. If this value is less than one, then the series ________ within a certain ________.

🤔 Part C: Critical Thinking

Explain, in your own words, why it is important to determine the radius of convergence when working with power series. Provide an example of what might happen if you ignore the radius of convergence.