Infinite Geometric Series Worksheets for High School Pre-Calculus

♾️ Topic Summary

An infinite geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. The series converges (has a finite sum) if the absolute value of the common ratio, $r$, is less than 1 (i.e., $|r| < 1$). The sum, $S$, of a convergent infinite geometric series can be found using the formula: $S = \frac{a}{1 - r}$, where $a$ is the first term of the series.

If $|r| \ge 1$, the series diverges and does not have a finite sum. Understanding when a series converges and how to calculate its sum is crucial in various applications, including calculus and physics. Let's test your knowledge!

🧮 Part A: Vocabulary

Match the terms with their definitions:

1. Term: A) A series with an infinite number of terms.
2. Common Ratio: B) The first number in a sequence.
3. Infinite Series: C) A sequence where each term is found by multiplying the previous term by a constant.
4. First Term: D) Each element in a sequence or series.
5. Geometric Sequence: E) The constant value multiplied by each term in a geometric sequence to get the next term.

✍️ Part B: Fill in the Blanks

An infinite geometric series ______ if the absolute value of the common ratio is less than 1. The formula to find the sum of a ______ infinite geometric series is $S = \frac{a}{1 - r}$, where $a$ is the ______ and $r$ is the common ______. If the absolute value of the common ratio is greater than or equal to 1, the series ______.

Possible answers: diverges, converges, ratio, first term, convergent

🤔 Part C: Critical Thinking

Explain, in your own words, why an infinite geometric series with a common ratio whose absolute value is greater than or equal to 1 does not have a finite sum. Provide an example to illustrate your explanation.