Absolute and Conditional Convergence

In mathematics, understanding the behavior of infinite series is crucial, and two important concepts in this context are absolute convergence and conditional convergence.

An infinite series is said to be absolutely convergent if the series formed by taking the absolute values of its terms also converges. On the other hand, a series is conditionally convergent if it converges, but it does not converge absolutely.

In this article, we will discuss about both these convergence types of series including tests to check. We will also discuss the key differences between Absolute and Conditional Convergence.

What is Absolute Convergence?

Absolute convergence is a concept in the study of infinite series in mathematics. An infinite series ∑a_n​ is said to converge absolutely if the series of absolute values ∑∣a_n∣ converges.

In other words, if you take the absolute value of each term in the series and sum them up, and the result is a finite number, then the original series is absolutely convergent.

Consider the series: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}

\sum_{n=1}^\infty \left|\frac{(-1)^{n+1}}{n^2}\right| = \sum_{n=1}^\infty \frac{1}{n^2}

The series \sum_{n=1}^\infty \frac{1}{n^2} is a p-series with p = 2, which is known to converge. Therefore, the original series converges absolutely.

Test for Absolute Convergence

There are many test to check absolute convergence, some of these are:

• Comparison Test
• Ratio Test

Let's discuss about these in detail as follows:

Comparison Test

The Comparison Test involves comparing the absolute value of the series in question with another series known to converge or diverge.

• Direct Comparison Test: If 0 ≤ ∣a_n∣ ≤ b_n​ for all n beyond some index, and ∑b_n​ converges, then ∑∣a_n∣ also converges, which implies that ∑a_n​ converges absolutely.
• Limit Comparison Test: If lim⁡_n→∞∣a_n∣/b_n = c where c is a positive finite number, and ∑b_n​ converges, then ∑∣a_n∣ also converges, indicating absolute convergence.

Ratio Test

The Ratio Test is useful when the terms of the series involve factorials or exponential functions. It is defined as follows:

L = lim_⁡n→∞∣a_n+1/a_n∣​​​

• If L < 1, the series ∑a_n​ converges absolutely.
• If L > 1 or L = ∞, the series ∑a_n diverges.
• If L = 1, the test is inconclusive.

Note: Some other tests are root test and integral test.

What is Conditional Convergence?

Conditional convergence occurs in an infinite series when the series converges, but it does not converge absolutely.

In other words, the series ∑a_n​ converges, but the series formed by taking the absolute values of its terms, ∑∣a_n​∣, diverges.

The classic example of a conditionally convergent series is the alternating harmonic series:

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

This series converges to ln(2). However, if you take the absolute values of the terms, you get the harmonic series:

\sum_{n=1}^{\infty} \left|\frac{(-1)^{n+1}}{n}\right| = \sum_{n=1}^{\infty} \frac{1}{n}

The harmonic series is known to diverge, hence the alternating harmonic series converges conditionally.

Test for Conditional Convergence

• Alternating Series Test (Leibniz's Test)
• Dirichlet’s Test

Alternating Series Test

The Alternating Series Test is used specifically for alternating series of the form ∑(−1)ⁿa_n​, where a_n​ is a positive, decreasing sequence that approaches zero.

Conditions for the test:

• a_n​ > 0
• a_n+1​ ≤ a_n​ (the terms a_n​​ are monotonically decreasing)
• lim_n→∞​a_n ​= 0

If these conditions are met, the series ∑(−1)ⁿa_n​ converges.

Dirichlet’s Test

Dirichlet’s Test can be applied to a series of the form ∑a_n​b_n​, where:

• a_n​​ is a sequence of complex numbers such that ∑a_n​​ is bounded.
• b_n​ is a monotonically decreasing sequence of positive real numbers tending to zero.

If these conditions are satisfied, the series ∑a_n​b_n converges.

Absolute Vs Conditional Convergence

The key differences between Absolute Convergence and Conditional Convergence are listed in the following table:

Conclusion

In conclusion, understanding absolute and conditional convergence is essential for anyone studying infinite series in mathematics. Absolute convergence occurs when the series of absolute values converges, providing a stronger form of convergence that ensures the sum remains the same regardless of term rearrangement. This makes absolutely convergent series robust and predictable.

Read More,

• Sequence and Series
• Sequence and Series Formulas
• Radius of Convergence

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Article Tags :

• Engineering Mathematics
• Mathematics
• Algebra