Determining Absolute or Conditional Convergence
Introduced| English |
|---|
| Absolute convergence/ˈæbsəluːt kənˈvɜːdʒəns/ |
| Conditional convergence/kənˈdɪʃənl kənˈvɜːdʒəns/ |
Signs alone do not settle convergence
- Two error corrections have terms (-1)^n/n and (-1)^n/n². Both alternate, but only the second is absolutely convergent.
- For 1/n, the absolute-value series is harmonic and diverges; the signed series passes the alternating-series test. For 1/n², the absolute-value p-series converges.
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.
Carry the reasoning to a new case
- Try (-1)^n n/(n+1).
- Its terms do not approach zero, so it diverges before any classification as conditional is possible.
Match each series to its convergence classification.
Check absolute values and the original terms. Alternation alone establishes neither convergence nor conditional convergence.
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.