The Ratio Test and Root Test are powerful tools for determining absolute convergence, especially for series involving factorials, exponentials, and powers. They are particularly useful because they don't require a comparison series.
🎯 In this section you will learn
- The Ratio Test and when to use it (factorials, exponentials)
- The Root Test and when to use it (powers of $n$, exponentials)
- How to interpret the results: $L < 1$ converges absolutely, $L > 1$ diverges, $L = 1$ inconclusive
📌 Ratio Test
$$ \text{Let } L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
$$ \text{If } L < 1, \text{ the series converges absolutely.} $$
$$ \text{If } L > 1, \text{ the series diverges.} $$
$$ \text{If } L = 1, \text{ the test is inconclusive.} $$
💡 When to use Ratio Test
Best for series with factorials ($n!$) or exponentials ($r^n$) because cancellation often occurs.
Test $\sum_{n=1}^\infty \frac{2^n}{n!}$ for convergence.
①
Set up the ratio
$\frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \frac{2}{n+1}$
②
Take the limit
$L = \lim_{n \to \infty} \frac{2}{n+1} = 0$
③
Conclusion
Since $L = 0 < 1$, the series converges absolutely.
Test $\sum_{n=1}^\infty \frac{n!}{2^n}$ for convergence.
- $\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{2^{n+1}} \cdot \frac{2^n}{n!} = \frac{n+1}{2}$
- $\lim_{n \to \infty} \frac{n+1}{2} = \infty$ (or $L = \infty > 1$)
- The series diverges.
Test $\sum_{n=1}^\infty \frac{1}{n^2}$ using the Ratio Test.
- $\frac{a_{n+1}}{a_n} = \frac{1/(n+1)^2}{1/n^2} = \frac{n^2}{(n+1)^2} \to 1$
- $L = 1$, test inconclusive (but we know it converges by $p$-series).
📌 Root Test
$$ \text{Let } L = \lim_{n \to \infty} |a_n|^{1/n} $$
$$ \text{If } L < 1, \text{ the series converges absolutely.} $$
$$ \text{If } L > 1, \text{ the series diverges.} $$
$$ \text{If } L = 1, \text{ the test is inconclusive.} $$
💡 When to use Root Test
Best for series where the $n$th term is raised to the $n$th power, like $(something)^n$.
Test $\sum_{n=1}^\infty \left(\frac{2n+3}{3n+2}\right)^n$ for convergence.
①
Compute $|a_n|^{1/n}$
$|a_n|^{1/n} = \frac{2n+3}{3n+2}$
②
Take the limit
$L = \lim_{n \to \infty} \frac{2n+3}{3n+2} = \frac{2}{3}$
③
Conclusion
Since $L = 2/3 < 1$, the series converges absolutely.
Test $\sum_{n=1}^\infty \frac{3^n}{n^n}$ for convergence.
- $|a_n|^{1/n} = \frac{3}{n}$
- $\lim_{n \to \infty} \frac{3}{n} = 0 < 1$ → converges absolutely.
⚠️ Important Notes
- Both tests give absolute convergence (if $L < 1$).
- If $L = 1$, the test is inconclusive — use another test.
- Ratio Test is usually better for factorials; Root Test is often better for terms like $(f(n))^n$.
🔍 Key Takeaways
- Ratio Test: $L = \lim |a_{n+1}/a_n|$. Converges if $L < 1$, diverges if $L > 1$.
- Root Test: $L = \lim |a_n|^{1/n}$. Converges if $L < 1$, diverges if $L > 1$.
- If $L = 1$, the test is inconclusive.
- Use Ratio Test for factorials and exponentials; Root Test for terms raised to the $n$th power.
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