Ratio and root tests

In the previous lecture, we learned the concept of "absolute convergence", which is a stronger type of convergence that guarantees many nice properties. Based on the observations on geometric series, we have two tests that can test for absolute convergence.

Tianran Chen

Department of Mathematics
Auburn University at Montgomery

Ratio and root tests

Both of these tests can identify series that can be approximated sufficiently well by geometric series, which we now fully understand.

Motivation: geometric series

Series that are almost geometric series

Ratio test

Let $\sum_{k=1}^\infty a_k$ be a series with $a_k \ne 0$ and $r = \lim_{k \to \infty} | \frac{a_{n+1}}{a_n} |$.

If $0 \le r \le 1$, then $\sum_{k=1}^\infty a_k$ converges absolutely.
If $r > 1$ or is infinite, then $\sum_{k=1}^\infty a_k$ diverges.

(If $r = 1$, then this test is inconclusive.)

Root test

The "root test" works in a slight different way, but the two essentially detect the same thing: whether or not a series can be approximated by a geometric series sufficiently well.

Let $\sum_{k=1}^\infty a_k$ be a series and $r = \lim_{k \to \infty} \sqrt[k]{ | a_k | }$.

If $0 \le r \le 1$, then $\sum_{k=1}^\infty a_k$ converges absolutely.
If $r > 1$ or is infinite, then $\sum_{k=1}^\infty a_k$ diverges.

(Again, if $r = 1$, then this test is inconclusive.)

Note the structural similarity between the two tests. This is expected as the two are designed to detect the same property.

Exercises

Using ratio and root tests, determine which of the following series is/are absolutely convergent

\[ \sum_{k=1}^\infty \frac{3}{k!} \]
\[ \sum_{k=2}^\infty ??? \]
\[ \sum_{k=2}^\infty ??? \]
\[ \sum_{k=2}^\infty ??? \]
\[ \sum_{k=2}^\infty ??? \]