Root test

Definition

The root test is a method used to determine the convergence or divergence of an infinite series by examining the nth root of the absolute value of its terms. It provides a useful criterion especially when dealing with series where ratio tests are inconclusive.

5 Must Know Facts For Your Next Test

1. The root test involves calculating $\lim_{{n \to \infty}} \sqrt[n]{{|a_n|}}$, where $a_n$ are the terms of the series.
2. If $L < 1$, where $L$ is the limit found using the root test, then the series converges absolutely.
3. If $L > 1$ or if $L$ is infinite, then the series diverges.
4. If $L = 1$, the root test is inconclusive, and other methods must be employed to determine convergence or divergence.
5. The root test can be particularly effective for series involving exponential functions or factorials.

Review Questions

• What does it mean if $\lim_{{n \to \infty}} \sqrt[n]{{|a_n|}} = L$ and $L < 1$?
• How does the root test determine divergence in an infinite series?
• What should you do if applying the root test gives you a limit of exactly one?

Related terms

Ratio Test: A method used to determine convergence or divergence of an infinite series by examining the limit of the ratio of successive terms.

Absolute Convergence: A concept where a series converges even when all its terms are replaced by their absolute values.

Infinite Series: A sum of infinitely many terms, typically expressed as a sequence whose partial sums approach a finite limit.