The Root Test is a method used in calculus to determine the convergence or divergence of an infinite series. Similar to the Ratio Test, the Root Test examines the limit of the series’ terms, but in this case, it involves the nth root of the absolute value of the terms.
Given an infinite series ∑a_n, the Root Test states:
- If lim (n→∞) |a_n|^(1/n) = L < 1, the series converges absolutely.
- If lim (n→∞) |a_n|^(1/n) = L > 1, the series diverges.
- If lim (n→∞) |a_n|^(1/n) = L = 1, the test is inconclusive, and the series may converge or diverge.
The Root Test is particularly useful for series with terms that involve exponential or polynomial functions. It is often used in conjunction with other convergence tests, such as the Ratio Test or the Comparison Test, to determine the behavior of an infinite series.