Root Test

Equation
$$\lim_{n \rightarrow oo} (a_{n})^{\frac{1}{n}}$$

Use
a. If $$n < 1$$ the series converges absolutely.
 * 1) 1)       Solve the limit as shown above and call its solution $$n$$
 * 2) 2)       Once you have derived $$n$$ follow these guidelines to determine convergence.

b. If $$n > 1$$ the series is divergent.

c. If $$n = 1$$ the test was inconclusive.

Explanation
The root test uses the Direct Comparison Test to state that if for all $$n \leq N$$ where N is some fixed natural number, then $$a_{n}^{\frac{1}{n}} < k < 1 $$ and therefore $$a_n < k^n < 1$$. Since the series $$k^n$$ converges through the Geometric Series Test so does $$a_n$$ through the user of the Direct Comparison Test.

Video Explanation


All Video Explanations by PatrickJMT