What is ratio test series?

The ratio test is a tool used in calculus to determine the convergence or divergence of an infinite series. It states that if the limit of the absolute value of the ratio of consecutive terms of a series is less than 1, then the series converges. If the limit is greater than 1 or equals infinity, then the series diverges. If the limit is equal to 1, the test is inconclusive.

Mathematically, the ratio test can be stated as follows:

Given a series ∑an, if Lim |an+1/an| as n approaches infinity is equal to L, then:

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

The ratio test is particularly useful for series that contain factorials or exponential terms, as it provides a simple method for determining convergence. It is a powerful tool in calculus and is often used in combination with other convergence tests to determine the convergence or divergence of a series.