Ratio and Root Test Analyzer — Theory
1. Purpose of the ratio and root tests
The ratio test and root test are used to study convergence of infinite series:
\[
\sum_{n=n_0}^{\infty} a_n.
\]
Both tests are especially useful for terms involving powers, factorials, and exponentials.
2. Absolute convergence
The ratio and root tests usually test absolute convergence. This means they apply to
\[
\sum_{n=n_0}^{\infty}|a_n|.
\]
If \(\sum |a_n|\) converges, then \(\sum a_n\) also converges.
3. Ratio test
The ratio test uses the limit
\[
L_R=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.
\]
The decision rule is:
\[
\begin{cases}
L_R<1, & \text{the series converges absolutely},\\
L_R>1, & \text{the series diverges},\\
L_R=1, & \text{the test is inconclusive}.
\end{cases}
\]
4. Root test
The root test uses the limit
\[
L_Q=\lim_{n\to\infty}\sqrt[n]{|a_n|}.
\]
The decision rule is:
\[
\begin{cases}
L_Q<1, & \text{the series converges absolutely},\\
L_Q>1, & \text{the series diverges},\\
L_Q=1, & \text{the test is inconclusive}.
\end{cases}
\]
5. Why factorials often work well with the ratio test
Factorials simplify when consecutive terms are divided.
\[
\frac{(n+1)!}{n!}=n+1.
\]
This cancellation makes the ratio test very useful for terms containing \(n!\).
6. Example: \(\sum \dfrac{n!}{n^n}\)
Let
\[
a_n=\frac{n!}{n^n}.
\]
Then
\[
\begin{aligned}
\left|\frac{a_{n+1}}{a_n}\right|
&=
\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{n^n}{n!}\\
&=
\left(\frac{n}{n+1}\right)^n.
\end{aligned}
\]
Therefore
\[
L_R=\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n=e^{-1}<1.
\]
So the series converges absolutely.
7. Example where the tests are inconclusive
For
\[
a_n=\frac{1}{n^2},
\]
the ratio test gives
\[
L_R=1.
\]
The root test also gives
\[
L_Q=1.
\]
Both tests are inconclusive, even though the \(p\)-series \(\sum 1/n^2\) converges.
8. Example of divergence
If
\[
a_n=\frac{n!}{5^n},
\]
then the ratio is approximately
\[
\left|\frac{a_{n+1}}{a_n}\right|
=
\frac{n+1}{5}.
\]
This grows without bound, so the series diverges.
9. When to prefer the root test
The root test is often convenient when \(a_n\) contains an \(n\)-th power:
\[
a_n=(\text{expression})^n.
\]
Taking the \(n\)-th root can simplify the expression quickly.
10. What the graph shows
The graph shows the computed sequences:
\[
R_n=\left|\frac{a_{n+1}}{a_n}\right|,
\qquad
Q_n=\sqrt[n]{|a_n|}.
\]
The horizontal line \(y=1\) is the threshold. Values clearly below \(1\) suggest absolute convergence,
while values clearly above \(1\) suggest divergence.
12. Common mistakes
- Forgetting absolute values: both tests use \(|a_n|\).
- Declaring divergence when \(L=1\): when the limit is \(1\), the test is inconclusive.
- Assuming inconclusive means divergent: inconclusive means another test is needed.
- Using too few terms numerically: early values may not represent the limiting behavior.
- Ignoring factorial growth: factorials can grow very quickly, so large \(N\) may cause overflow.
