1. What is the Series Ratio?

The series ratio (r) is the ratio between consecutive terms in a sequence or series. It provides the basic ratio for analyzing series behavior and is fundamental in determining whether a series converges or diverges.

2. How Does the Calculator Work?

The calculator uses the series ratio formula:

Where:

• \( r \) — Series ratio (dimensionless)
• \( a_{n+1} \) — Next term value
• \( a_n \) — Current term value

Explanation: The ratio compares the magnitude of consecutive terms in a sequence, which is particularly useful in analyzing geometric series and power series.

3. Importance of Series Ratio

Details: The series ratio is crucial for determining convergence of infinite series, analyzing growth rates, and understanding the behavior of sequences in mathematical analysis.

4. Using the Calculator

Tips: Enter the values of two consecutive terms in the series. The denominator (current term) must be non-zero for the calculation to be valid.

5. Frequently Asked Questions (FAQ)

Q1: What does the ratio tell us about a series?
A: The ratio helps determine if a series is geometric, and if so, what the common ratio is. It's also used in the ratio test for convergence.

Q2: What if the ratio approaches 1?
A: If |r| approaches 1, the ratio test is inconclusive, and other convergence tests must be used.

Q3: Can the ratio be negative?
A: Yes, the ratio can be negative if consecutive terms have opposite signs, indicating an alternating series.

Q4: How is this different from the common ratio?
A: For geometric series, the series ratio equals the common ratio. For non-geometric series, the ratio may vary between terms.

Q5: What's the relationship with the ratio test?
A: The ratio test uses the limit of the series ratio to determine absolute convergence (if limit < 1) or divergence (if limit > 1).