Math Lesson 12.4.7 - The Root Convergence Test

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Welcome to our Math lesson on The Root Convergence Test, this is the seventh lesson of our suite of math lessons covering the topic of Infinite Series Explained, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Root Convergence Test Explained

This test is particularly important in cases when the ratio test cannot provide an answer for the convergence of an infinite series. Let's explain what the root convergence test consists of.

Let Σa_n be an infinite series with non-negative terms and suppose that

where L is a finite number (it represents the limit of the series).

In this case, we face with three possible options:

1. If L < 0, then Σa_n is convergent;
2. If L > 0, then Σa_n is divergent;
3. If L = 1, then the test is conclusive, i.e. it cannot give us an exact answer about the convergence of the given series Σa_n.

For example, the series

is convergent, because

On the other hand, the series

is divergent, as

Example 5

Check the convergence of the following series

1. S₁ = ∞∑_{n = 1}(3n - 4)ⁿ
2. S₂ = ∞∑_{n = 1}2/4 + 3nⁿ

Solution 5

1. We have
   limn → ∞⁡(3n - 4)^n1/n = limn → ∞⁡(3n - 4)
   = 3 ∙ ∞-4
   = ∞
   Since ∞ > 1, the series S₁ is divergent.
2. We have
   limn → ∞⁡2/4 + 3n^n1/n = limn → ∞⁡2/4 + 3n
   = 2/4 + 3 ∙ ∞
   = 2/∞
   = 0
   Since 0 < 1, then the series S₂ is convergent.

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