Kuzee
Infinite Series

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.

SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
∣an∣\lvert a_n\rvertMathematical symbolAbsolute valueThe absolute value of term ana_n

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}.

series convergenceabsolute convergenceconditional convergencealternating series

Approach: First determine absolute convergence; if absolutely convergent, then the original series converges.

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 Used in This Article

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