1. What Is Convergence Vs Divergence?

In mathematics, a series is convergent if the sum of its terms approaches a finite value as more terms are added. A series is divergent if the sum does not approach any finite limit.

2. How Does The Calculator Work?

The calculator evaluates the partial sum of the series and applies basic convergence tests:

Where:

• \( a_n \) — The sequence expression (e.g., 1/n, 1/n^2)
• \( a \) — Lower limit of summation
• \( b \) — Upper limit of summation

Explanation: The calculator computes a partial sum and checks if the terms are decreasing to determine convergence.

3. Importance Of Series Convergence

Details: Determining convergence is fundamental in calculus and analysis, with applications in physics, engineering, and probability theory.

4. Using The Calculator

Tips: Enter the sequence expression using standard mathematical notation. Use 'n' as the variable. Set appropriate summation limits.

5. Frequently Asked Questions (FAQ)

Q1: What are some common convergent series?
A: Geometric series with |r| < 1, p-series with p > 1, and alternating series that meet certain conditions.

Q2: What are some common divergent series?
A: Harmonic series (1/n), geometric series with |r| ≥ 1, and p-series with p ≤ 1.

Q3: What is the difference between sequence and series?
A: A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.

Q4: What are the main convergence tests?
A: Ratio test, root test, integral test, comparison test, and alternating series test.

Q5: Can a series converge conditionally?
A: Yes, a series may converge conditionally if it converges but doesn't converge absolutely.