Summation Convergence Calculator

Ratio Test Formula:

\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \]

Convergence Result:

Ratio Limit Value:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Is The Ratio Test For Series Convergence?

The ratio test is a method used to determine the convergence or divergence of an infinite series. It examines the limit of the ratio of consecutive terms in the series to make this determination.

2. How Does The Ratio Test Work?

The ratio test uses the formula:

\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \]

Where:

If L < 1, the series converges absolutely
If L > 1, the series diverges
If L = 1, the test is inconclusive

Explanation: The test compares the growth rate of consecutive terms. If terms eventually decrease rapidly enough (ratio < 1), the series converges.

3. Importance Of Convergence Testing

Details: Determining whether an infinite series converges is fundamental in mathematical analysis, with applications in physics, engineering, and many areas of mathematics where infinite sums are used to represent functions or solve problems.

4. Using The Calculator

Tips: Enter the general term of your series using standard mathematical notation. Use 'n' as your variable unless specified otherwise. The calculator will compute the limit of the ratio of consecutive terms.

5. Frequently Asked Questions (FAQ)

Q1: When should I use the ratio test?
A: The ratio test is particularly useful for series containing factorials, exponential functions, or other terms where the ratio of consecutive terms simplifies nicely.

Q2: What if the limit equals exactly 1?
A: When L = 1, the ratio test is inconclusive. You'll need to use another convergence test such as the comparison test, integral test, or root test.

Q3: Can the ratio test determine absolute convergence?
A: Yes, the ratio test can establish absolute convergence when L < 1, which is stronger than regular convergence.

Q4: Are there series where the ratio test fails?
A: Yes, for series where the limit doesn't exist or equals 1, the ratio test cannot determine convergence. Some alternating series and p-series may require different tests.

Q5: How accurate is this calculator?
A: This calculator provides a mathematical analysis based on the ratio test principle. For complex series, manual verification may be recommended.