📚 Topic Summary
An infinite geometric series is the sum of an infinite number of terms where each term is multiplied by a constant ratio to get the next term. The key to determining if it converges (has a finite sum) is the common ratio, $r$. If the absolute value of $r$ is less than 1 (i.e., $|r| < 1$), the series converges, and we can find its sum using the formula: $S = \frac{a}{1-r}$, where $a$ is the first term. If $|r| \ge 1$, the series diverges, meaning it does not have a finite sum.
This worksheet provides practice in identifying geometric series, determining convergence, and calculating the sum of convergent infinite geometric series.
🧮 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Common Ratio |
A. The value that each term is multiplied by to get the next term. |
| 2. Convergent Series |
B. An infinite series that has a finite sum. |
| 3. Divergent Series |
C. An infinite series that does not have a finite sum. |
| 4. Infinite Geometric Series |
D. The sum of an infinite number of terms where each term is multiplied by a common ratio. |
| 5. First Term |
E. The initial value in a geometric sequence or series. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words.
An infinite geometric series _________ if the absolute value of the common ratio is less than 1. The sum of a convergent infinite geometric series is given by the formula $S = \frac{a}{1-r}$, where $a$ is the _________ term and $r$ is the _________ _________. If the absolute value of the common ratio is greater than or equal to 1, the series _________.
🤔 Part C: Critical Thinking
Explain, in your own words, why the common ratio must have an absolute value less than 1 for an infinite geometric series to converge. Provide an example to support your explanation.