Determining Absolute or Conditional Convergence · Determinando Convergência Absoluta ou Condicional
| English | Português |
|---|---|
| Absolute convergence/ˈæbsəluːt kənˈvɜːdʒəns/ | convergência absoluta |
| Conditional convergence/kənˈdɪʃənl kənˈvɜːdʒəns/ | convergência condicional |
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 · forte 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 · Convergência com inversão de sinal
An alternating series may converge only because of cancellation — its absolute-value series can still diverge (conditional). · Uma série alternada pode convergir apenas devido à cancelamento — sua série de valor absoluto ainda pode divergir (condicional).
A series is absolutely convergent when... · Uma série é absolutamente convergente quando...
Absolute = the absolute-value series converges. · Absoluta = a série de valor absoluto converge.
Absolute convergence implies (ordinary) convergence. · A convergência absoluta implica (convergência) ordinária.
The strong kind always converges. · O tipo forte sempre converge.
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 $\sum|a_n|$... · Uma série é condicionalmente convergente quando $\sum a_n$ converge, mas $\sum|a_n|$...
Converges, but absolute-value series diverges. · Converge, mas a série de valor absoluto diverge.
Rearranging the terms of a conditionally convergent series can change its sum. · Reorganizar os termos de uma série condicionalmente convergente pode alterar sua soma.
Only absolutely convergent series rearrange safely. · Apenas séries absolutamente convergentes podem ser reorganizadas com segurança.
The decision procedure
- 1. Test · Teste $\sum|a_n|$. 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... · A série harmônica alternada $\sum\tfrac{(-1)^n}{n}$ é...
Converges, but $\sum\tfrac1n$ diverges → conditional. · Converge, mas $\sum\tfrac1n$ diverge → condicional.
To classify, you first test... · Para classificar, você primeiramente testa...
Test the absolute-value series first. · Teste a série de valor absoluto primeiro.
Test the absolute-value series · série $\sum|a_n|$ first. Absolute = $\sum|a_n|$ converges; conditional = $\sum a_n$ converges but $\sum|a_n|$ diverges. A conditionally convergent series is not · não 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 · se $\sum|a_n|$ converges (the strong kind — implies convergence, allows rearrangement). It is conditionally convergent if · se $\sum a_n$ converges but $\sum|a_n|$ diverges (relies on sign cancellation). Test $\sum|a_n|$ first, then the signed series.