root test- sequences and seriesthis is the final

Question

Root Test- Sequences and Series

This is the final test for series convergence that we're going to be searching for at.  Like with the Ratio Test this test will as well tell whether a series is absolutely convergent or not rather as compared to simple convergence.

Assume that we have the series ∑a_n Define,

Then,

a. if L < 1  the series is absolutely convergent and therefore convergent).

b. if L > 1  the series is divergent.

c. if L =  1 the series may be divergent, may be conditionally convergent, or absolutely convergent.

Since with the ratio test, if we get L= 1 the root test will tell us nothing and we'll need to utilize another test to find out the convergence of the series.  As well note that if L=1 in the Ratio Test afterwards the Root Test will as well give L= 1.