Determine whether the geometric series is convergent or divergent.
Understand the Problem
The question is asking us to analyze a geometric series to determine if it converges to a limit or diverges to infinity. To solve this, we will need to identify the common ratio of the series and apply the criteria for convergence of geometric series.
Answer
The series converges if $|r| < 1$, and diverges if $|r| \geq 1$.
Answer for screen readers
The series converges if $|r| < 1$, and diverges if $|r| \geq 1$.
Steps to Solve
- Identify the first term and common ratio
For a geometric series, the first term is often denoted as $a$, and the common ratio is denoted as $r$.
- Criteria for convergence
A geometric series converges if the absolute value of the common ratio is less than 1 ($|r| < 1$). If $|r| \geq 1$, the series diverges.
- Applying the criteria
Determine the absolute value of the common ratio you identified in step 1. If it is less than 1, conclude that the series converges; otherwise, conclude that it diverges.
- Conclusion
Based on the value of $|r|$, state your conclusion regarding the convergence or divergence of the series.
The series converges if $|r| < 1$, and diverges if $|r| \geq 1$.
More Information
A geometric series is a series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding convergence helps in many fields, including physics and engineering, where infinite sums are applied.
Tips
- Mistaking the common ratio: Ensure you identify the correct ratio between successive terms.
- Failing to take the absolute value of the common ratio before applying the convergence criteria.
