Determining Absolute or Conditional Convergence
| English | Chinese | Pinyin |
|---|---|---|
| Absolute convergence | 绝对收敛 | jué duì shōu liǎn |
| Conditional convergence | 条件收敛 | tiáo jiàn shōu liǎn |
Two grades of convergence
- A convergent series can converge in a strong way or a fragile way.
- Absolute convergence 绝对收敛: the series of absolute values $\sum|a_n|$ also converges.
- Conditional convergence 条件收敛: the series converges, but $\sum|a_n|$ diverges — it relies on cancellation.
- Distinguishing them tells you how robust the sum is.
Absolute convergence is stronger
- Check $\sum|a_n|$ first. If it converges, the original series converges absolutely.
- Absolute convergence implies ordinary convergence — it's the safe, sturdy kind.
- You can even rearrange an absolutely convergent series freely without changing the sum.
- Most convergence tests (ratio, comparison) actually test absolute convergence.
Convergence with sign flips
An alternating series may converge only because of cancellation — its absolute-value series can still diverge (conditional).
A series is absolutely convergent when...
Absolute = the absolute-value series converges.
Absolute convergence implies (ordinary) convergence.
The strong kind always converges.
Conditional convergence relies on the signs
- If $\sum a_n$ converges but $\sum|a_n|$ diverges, the convergence is conditional.
- The alternating harmonic series $\sum\tfrac{(-1)^n}{n}$ is the classic case: it converges, but $\sum\tfrac1n$ diverges.
- Its convergence depends entirely on the sign flips cancelling — remove them and it blows up.
- Fragile, but still convergent.
A series is conditionally convergent when $\sum a_n$ converges but $\sum|a_n|$...
Converges, but absolute-value series diverges.
Rearranging the terms of a conditionally convergent series can change its sum.
Only absolutely convergent series rearrange safely.
The decision procedure
- 1. Test $\sum|a_n|$. If it converges → absolutely convergent (done).
- 2. If $\sum|a_n|$ diverges, test $\sum a_n$ itself (often the Alternating Series Test).
- If $\sum a_n$ converges → conditionally convergent; if not → divergent.
- Absolute value first, then the signed series.
The alternating harmonic series $\sum\tfrac{(-1)^n}{n}$ is...
Converges, but $\sum\tfrac1n$ diverges → conditional.
To classify, you first test...
Test the absolute-value series first.
Test the absolute-value series $\sum|a_n|$ first. Absolute = $\sum|a_n|$ converges; conditional = $\sum a_n$ converges but $\sum|a_n|$ diverges. A conditionally convergent series is not the same as absolutely convergent — its sum can even change if you rearrange the terms. Don't call a merely-convergent alternating series "absolutely" convergent.
Classify $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n}$.
- Absolute values: $\sum\tfrac1n$ is the harmonic series → diverges. So not absolutely convergent.
- The series itself: alternating, $b_n=\tfrac1n$ decreasing to $0$ → converges (Alternating Series Test).
- Converges but not absolutely → conditionally convergent.
A series is absolutely convergent if $\sum|a_n|$ converges (the strong kind — implies convergence, allows rearrangement). It is conditionally convergent if $\sum a_n$ converges but $\sum|a_n|$ diverges (relies on sign cancellation). Test $\sum|a_n|$ first, then the signed series.