Nth Term Test Calculator

Historical Background

The Nth Term Test, also known as the Divergence Test, is one of the simplest tools used in determining the behavior of an infinite series. It dates back to early developments in calculus and series theory, where mathematicians sought to understand the convergence or divergence of infinite sums. The test provides a quick and easy method for determining if a series definitely diverges.

Calculation Formula

The Nth Term Test states that if the limit of the nth term of a series does not approach zero as \( n \to \infty \), the series diverges. The formula is:

\[ \lim_{{n \to \infty}} a_n \]

If this limit is not zero, the series diverges. If the limit is zero, the test is inconclusive, and further methods are needed.

Example Calculation

Consider the series \( a_n = \frac{1}{n} \):

\[ \lim_{{n \to \infty}} \frac{1}{n} = 0 \]

In this case, the Nth Term Test is inconclusive, as the limit equals zero. Other tests like the comparison test or ratio test would be needed.

For \( a_n = n \), the limit is:

\[ \lim_{{n \to \infty}} n = \infty \]

Thus, the series diverges.

Importance and Usage Scenarios

The Nth Term Test is important because it provides a quick first step in determining whether an infinite series can converge or not. If the limit of the terms does not go to zero, there's no need for more complex tests—the series diverges. It's used in calculus, mathematical analysis, and many applications in physics and engineering where infinite series models are common.

Common FAQs

1. What does it mean if the Nth Term Test is inconclusive?

   • It means the series might still converge or diverge, and other tests must be applied to determine its behavior.

2. Can the Nth Term Test prove convergence?

   • No, the Nth Term Test can only prove divergence. If the test is inconclusive, other methods are required.

3. What happens if the limit of the nth term is not a number?

   • If the limit is infinity or does not exist, the series diverges.