Does the series converge or diverge: The series convergesdiverges
Use the Root Test to determine whether an infinite series converges or diverges.
Root Test
In this section we will determine whether an infinite series converges or diverges using the Root Test. The idea behind this test is to determine if the given series is behaving similarly to a geometric series. For a geometric series, the root of the term approaches as i.e.,
For a general series, if the limit of the root of the absolute value of the term approaches a value as , then the series behaves like a geometric series with ratio equal to . We state this precisely in the following theorem.
Root Test For the series let The series converges (absolutely) if and diverges if . If
then no conclusion can be made.
example 1 Determine whether the series converges or diverges:
By the Root Test, Since , the series converges.
Here is a detailed, lecture style video on the Root Test:
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