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Absolute and Conditional Convergence

Absolute Convergence

Definition of Absolute Convergence

If the series ∑n=1∞∣an∣\sum_{n=1}^{\infty} |a_n| converges, then the series ∑n=1∞an\sum_{n=1}^{\infty} a_n is said to converge absolutely.

Conditional Convergence

Definition of Conditional Convergence

If the series ∑n=1∞an\sum_{n=1}^{\infty} a_n converges, but the series ∑n=1∞∣an∣\sum_{n=1}^{\infty} |a_n| diverges, then the series ∑n=1∞an\sum_{n=1}^{\infty} a_n is said to converge conditionally.

符号说明
SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
$a_n$Mathematical symbol

Absolutely Convergent Series Must Converge

Absolute and Conditional Convergence
  • Absolute Convergence: If ∑n=1∞∣an∣\sum_{n=1}^{\infty} |a_n| converges, then ∑n=1∞an\sum_{n=1}^{\infty} a_n converges absolutely
  • Conditional Convergence: If ∑n=1∞an\sum_{n=1}^{\infty} a_n converges but ∑n=1∞∣an∣\sum_{n=1}^{\infty} |a_n| diverges, then ∑n=1∞an\sum_{n=1}^{\infty} a_n converges conditionally

Testing Strategy

  1. First determine the convergence of the absolute value series
  2. If absolutely convergent, then the original series converges
  3. If the absolute value series diverges, then determine if the original series converges conditionally

Examples

Example 1

Determine the convergence of the series ∑n=1∞(−1)nn2\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}.

Solution:

  1. Consider the absolute value series: ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}
  2. This is a p-series with p=2>1p = 2 > 1, so it converges absolutely
  3. Since it converges absolutely, the original series converges

Exercises

Exercise 1

Determine the convergence of the series ∑n=1∞(−1)nn2\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}.

Reference Answer(4 个标签)
series convergenceabsolute convergenceconditional convergencealternating series

Problem-solving approach: First determine absolute convergence; if absolutely convergent, then the original series converges.

Detailed steps:

  1. Consider the absolute value series: ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}
  2. This is a p-series with p=2>1p = 2 > 1, so it converges absolutely
  3. Since it converges absolutely, the original series converges

Answer: The series converges.

Summary

Symbols Appearing in This Article

SymbolTypePronunciation/DescriptionMeaning in This Article
ana_nMathematical symbolGeneral termThe nth term in the series

Chinese-English Glossary

Chinese TermEnglish TermIPADescription
绝对收敛absolute convergence/ˈæbsəluːt kənˈvɜːdʒəns/The case where the absolute value series converges
条件收敛conditional convergence/kənˈdɪʃənəl kənˈvɜːdʒəns/The case where the series converges but the absolute value series diverges
收敛convergence/kənˈvɜːdʒəns/The partial sum sequence has a finite limit
发散divergence/daɪˈvɜːdʒəns/The partial sum sequence has no finite limit
绝对值级数absolute value series/ˈæbsəluːt ˈvæljuː ˈsɪəriːz/The series formed by taking absolute values of each term
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