1. What is the Comparison Test?

The Comparison Test is a method used in calculus to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior.

2. How Does the Calculator Work?

The calculator uses the comparison test formula:

Where:

• \( a_n \) — The sequence being tested
• \( b_n \) — The comparison sequence with known behavior
• \( L \) — The limit of the ratio (must be positive and finite)

Explanation: If the limit L exists and is positive and finite, then both series either converge or diverge together.

3. Importance of Comparison Test

Details: The comparison test is fundamental in mathematical analysis for determining series convergence without directly evaluating the infinite sum.

4. Using the Calculator

Tips: Enter mathematical expressions for sequences a_n and b_n using standard mathematical notation. The calculator will compute the limit of their ratio.

5. Frequently Asked Questions (FAQ)

Q1: What types of sequences can be compared?
A: The comparison test works for positive-term sequences where both sequences have the same asymptotic behavior.

Q2: What if the limit L = 0?
A: If L = 0 and b_n converges, then a_n also converges. If L = ∞ and b_n diverges, then a_n also diverges.

Q3: Can the comparison test be used for alternating series?
A: No, the comparison test is specifically designed for series with non-negative terms.

Q4: What are common comparison sequences?
A: Common comparison sequences include p-series (1/n^p), geometric series, and other well-understood convergent/divergent series.

Q5: When should I use the limit comparison test vs the direct comparison test?
A: The limit comparison test is often easier to apply as it doesn't require establishing inequalities between terms.