When a series has both positive and negative terms (like alternating series), we need to distinguish between absolute convergence (the series of absolute values converges) and conditional convergence (the series converges, but the absolute series diverges). This distinction matters because absolutely convergent series behave nicely — they can be rearranged without changing the sum.
🎯 In this section you will learn
- The definition of absolute convergence and conditional convergence
- That absolute convergence implies convergence
- How to test for absolute convergence (use tests for positive series)
- Examples: alternating harmonic series (conditionally convergent) vs $p$-series with $p>1$ (absolutely convergent)
📌 Definitions
$$ \sum a_n \text{ converges } \textbf{absolutely} \text{ if } \sum |a_n| \text{ converges.} $$
$$ \sum a_n \text{ converges } \textbf{conditionally} \text{ if } \sum a_n \text{ converges but } \sum |a_n| \text{ diverges.} $$
💡 Key Theorem
If $\sum |a_n|$ converges, then $\sum a_n$ converges. Absolute convergence implies convergence.
Test $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}$ for absolute/conditional convergence.
①
Absolute series
$\sum |\frac{(-1)^{n-1}}{n^2}| = \sum \frac{1}{n^2}$
②
Check convergence
$\sum 1/n^2$ is a $p$-series with $p=2>1$, so it converges.
③
Conclusion
Since $\sum |a_n|$ converges, the series converges absolutely.
Test $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}$ for absolute/conditional convergence.
①
Absolute series
$\sum |\frac{(-1)^{n-1}}{n}| = \sum \frac{1}{n}$ (harmonic series)
②
Check convergence
$\sum 1/n$ diverges (harmonic series).
③
Check original series
By Alternating Series Test, $\sum (-1)^{n-1}/n$ converges.
④
Conclusion
The series converges but not absolutely → conditionally convergent.
Test $\sum_{n=1}^\infty \frac{(-1)^{n-1} n}{n+1}$ for absolute/conditional convergence.
- $\lim a_n = \pm 1 \neq 0$, so the series diverges (nth term test).
- Not absolutely convergent (diverges), not conditionally convergent (doesn't converge at all).
📌 Strategy for Determining Absolute/Conditional Convergence
①
Check $\lim a_n$
If $\lim a_n \neq 0$, the series diverges → done.
②
Test for absolute convergence
Consider $\sum |a_n|$; use ratio test, root test, comparison, etc. If it converges → absolutely convergent.
③
Test original series (if needed)
If $\sum |a_n|$ diverges, test $\sum a_n$ (e.g., with Alternating Series Test). If it converges → conditionally convergent.
Determine if $\sum_{n=1}^\infty \frac{(-1)^n n}{2^n}$ converges absolutely, conditionally, or diverges.
①
Test absolute series
$\sum \frac{n}{2^n}$ (positive terms). Use Ratio Test:
②
Apply Ratio Test
$\lim \frac{(n+1)/2^{n+1}}{n/2^n} = \lim \frac{n+1}{2n} = \frac{1}{2} < 1$ → converges absolutely.
③
Conclusion
Since $\sum |a_n|$ converges, the series converges absolutely.
⚠️ Important Notes
- Absolute convergence is stronger: Absolutely convergent series are "well-behaved" (rearrangement doesn't change sum).
- Conditionally convergent series: The alternating harmonic series is the classic example. Its sum changes if you rearrange terms!
- Most series on the AP exam are either absolutely convergent or divergent. Conditional convergence appears mainly with alternating $p$-series where $0 < p \le 1$.
🔍 Key Takeaways
- Absolutely convergent: $\sum |a_n|$ converges.
- Conditionally convergent: $\sum a_n$ converges, but $\sum |a_n|$ diverges.
- Absolute convergence ⇒ convergence.
- Test absolute convergence first — if it converges, you're done.
← Back to Module Page