1. What is the Ratio of Successive Terms?

The ratio of successive terms (r = a_n+1/a_n) is a fundamental concept in sequence analysis, used to test for convergence in infinite series and to identify geometric sequences.

2. How Does the Calculator Work?

The calculator uses the simple ratio formula:

Where:

• \( a_{n+1} \) — The next term in the sequence
• \( a_n \) — The current term in the sequence
• \( r \) — The ratio between terms (dimensionless)

Explanation: This ratio helps determine if a sequence is geometric (constant ratio) and is crucial in convergence tests for infinite series.

3. Importance of Ratio Calculation

Details: Calculating the ratio between consecutive terms is essential for:

• Determining if a sequence is geometric
• Applying the ratio test for series convergence
• Predicting future terms in a sequence
• Analyzing growth rates in mathematical models

4. Using the Calculator

Tips:

• Enter the value of the next term (a_n+1)
• Enter the value of the current term (a_n)
• The current term cannot be zero (would cause division by zero)
• The result is dimensionless (a pure number)

5. Frequently Asked Questions (FAQ)

Q1: What does the ratio tell us about a sequence?
A: A constant ratio indicates a geometric sequence. The value of the ratio helps determine convergence properties.

Q2: How is this used in convergence tests?
A: In the ratio test, if lim|r| < 1 the series converges absolutely, if > 1 it diverges, and if = 1 the test is inconclusive.

Q3: What if my current term is zero?
A: The ratio is undefined when the denominator (current term) is zero, as division by zero is mathematically undefined.

Q4: Can this be used for non-numerical sequences?
A: No, this calculator is designed for numerical sequences where division is defined.

Q5: What's the difference between ratio and common difference?
A: Ratio is for geometric sequences (multiplicative), common difference is for arithmetic sequences (additive).