Ratio and Root Tests

Ratio Test and Root Test

By the end of this lesson, you will be able to apply the Ratio Test and Root Test to determine absolute convergence of series.

The Ratio Test

For $\sum a_n$ (with $a_n > 0$ or $a_n$ nonzero), compute:

$$\rho = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$$

$\rho$	Conclusion
$\rho < 1$	Series converges absolutely
$\rho > 1$	Series diverges
$\rho = 1$	Inconclusive — use another test

Best for: series with factorials or exponentials (they telescope nicely in the ratio).

Example: $\sum \frac{2^n}{n!}$

$\frac{a_{n+1}}{a_n} = \frac{2^{n+1}/(n+1)!}{2^n/n!} = \frac{2}{n+1} \to 0$

$\rho = 0 < 1$ → converges absolutely.

Example: $\sum \frac{n!}{n^n}$

$\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n} = \frac{(n+1) \cdot n^n}{(n+1)^{n+1}} = \left(\frac{n}{n+1}\right)^n = \left(1-\frac{1}{n+1}\right)^n \to e^{-1} < 1$

→ converges.

The Root Test

$$\rho = \lim_{n\to\infty}\sqrt[n]{|a_n|}$$

Same conclusions as the Ratio Test.

Best for: series where $a_n = f(n)^n$ — the $n$th root simplifies nicely.

Example: $\sum \left(\frac{2n}{3n+1}\right)^n$

$\sqrt[n]{a_n} = \frac{2n}{3n+1} \to \frac{2}{3}$

$\rho = 2/3 < 1$ → converges.

When $\rho = 1$: These Tests Fail

$\sum 1/n$ ($\rho = 1$, diverges); $\sum 1/n^2$ ($\rho = 1$, converges). Both give $\rho = 1$, so the Ratio and Root tests can't distinguish them.

When $\rho = 1$: try p-series comparison, integral test, or alternating series test.

Summary

Ratio Test: $\rho = \lim|a_{n+1}/a_n|$; $< 1$ → converge; $> 1$ → diverge; $= 1$ → inconclusive.

Root Test: $\rho = \lim|a_n|^{1/n}$; same rules.

Use Ratio Test for: factorials, exponentials, products of sequences.

Use Root Test for: expressions raised to the $n$th power.

Both give $\rho = 1$: inconclusive — use another test.