Conditional convergence is convergence with a condition that is a series is said to be conditionally convergent if it converges, but not absolutely. This means that while the series ∑an converges, the series of the absolute values ∑∣an∣ diverges.
A classic example of a conditionally convergent series is the alternating harmonic series ∑(−1)n+11/n, which converges to ln(2), despite the fact that the harmonic series ∑1/n diverges. This divergence of the series of absolute values indicates that the positive and negative terms in the original series play a crucial role in its convergence.
In this article, we will discuss the concept of Conditional Convergence in detail including examples and various test to find weather series is conditionally convergent or not.
What is Conditional Convergence?
Conditional convergence is a term used in mathematics to describe a specific behavior of an infinite series and refers to a property of an infinite series in mathematics where the series converges, but the series formed by taking the absolute values of its terms diverges.
To be more specific, any conditionally convergent series converges to a finite limit only because of the specific arrangement and cancellation of its terms, despite the fact that the magnitude of the terms grows large without bound.
Definition of Conditional Convergence
A series \sum_{n=0}^{\infty} a_n is said to be conditionally convergent if:
- The series \sum_{n=0}^{\infty} a_nan converges to a finite value.
- The series of the absolute values of its terms,\sum_{n=0}^{\infty} |a_n|, diverges.
Examples of Conditionally Convergent Series
Some of the common examples of series that converges conditionally are:
Alternating Harmonic Series
The alternating harmonic series is a well-known example of a conditionally convergent series:
\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots
This series converges to ln(2), but the harmonic series \sum_{n=1}^{\infty} \frac{1}{n}, which consists of the absolute values of the terms, diverges.
Alternating Series with Logarithmic Terms
Another example is the series involving logarithmic terms:
\sum_{n=1}^{\infty} (-1)^n \frac{\ln(n)}{n}
This series converges conditionally. However, the series of absolute values \sum_{n=1}^{\infty} \frac{\ln(n)}{n} diverges.
Series Involving Trigonometric Functions
Series that include alternating trigonometric functions can also be conditionally convergent, such as:
\sum_{n=1}^{\infty} (-1)^n \frac{\cos(n)}{n}
This series converges conditionally, but the series \sum_{n=1}^{\infty} \frac{\cos(n)}{n} does not converge absolutely.
Alternating p-Series
For 0 < p ≤ 1, the alternating p-series:
\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n^p} is conditionally convergent.
For example, when p = 1, it reduces to the alternating harmonic series, which is conditionally convergent. For p < 1, the series still converges conditionally because the positive p-series \sum_{n=1}^{\infty} \frac{1}{n^p} diverges
Tests for Conditional Convergence
Some of the most common test for conditional convergence are:
- Alternating Series Test
- Dirichlet's Test
- Abel's Test
Alternating Series Test
The Alternating Series Test is used specifically for alternating series of the form \sum (-1)^n a_n, where an is a positive, decreasing sequence that approaches zero.
Conditions for the test:
- an > 0
- an+1 ≤ an (the terms an are monotonically decreasing)
- \lim_{n \to \infty} a_n = 0
If these conditions are met, the series ∑(−1)nan\sum (-1)^n a_n∑(−1)nan converges.
Dirichlet's Test
Dirichlet's Test can be applied to a series of the form \sum a_n b_n, where:
- an is a sequence of complex numbers such that ∑an is bounded.
- bn is a monotonically decreasing sequence of positive real numbers tending to zero.
If these conditions are satisfied, the series ∑anbn converges.
Abel's Test
Abel's Test is another method for determining the conditional convergence of series of the form ∑anbn, where:
- an is a sequence whose partial sums are bounded.
- bn is a monotonically decreasing sequence that converges to zero.
If these conditions are met, the series ∑anbn converges.
Absolute vs Conditional Convergence
The key difference between absolute and conditional convergence are listed in the following table:
| Feature | Absolute Convergence | Conditional Convergence |
|---|
| Definition | A series ∑an converges absolutely if:
∑|an| < ∞ | A series ∑an converges conditionally if:
The series ∑an converges.
The series ∑|an| converges. |
| Rearrangement Property | The series remains convergent and sums to the same value regardless of the order of terms. | The series can be rearranged to converge to different values or even diverge. |
| Example | Geometric series ∑(1/2)n. | Alternating harmonic series ∑(−1)n+1(1/n). |
| Common Tests/n | Ratio Test, Root Test, Comparison Test. | Alternating Series Test, Dirichlet's Test, Abel's Test. |
| Implication of Convergence | Implies both absolute and conditional convergence. | Does not imply absolute convergence. |
| Behavior of Terms | Terms decrease rapidly enough in magnitude. | Alternating terms with decreasing magnitude but not rapidly enough for absolute convergence. |
| Impact of Positive and Negative Terms | Positive and negative terms do not affect convergence as much. | Positive and negative terms significantly impact convergence. |
Conclusion
Conditional convergence is an essential concept in the study of infinite series in mathematics. It describes a series that converges to a finite value, even though the series of its absolute values diverges. This unique behavior highlights the importance of the arrangement of terms in a series and demonstrates that convergence can depend heavily on the specific sequence of positive and negative terms.
Read More,
Define Conditional Convergence.
Conditional convergence occurs when an infinite series ∑an converges, but the series of its absolute values ∑∣an∣ diverges.
How is Conditional Convergence different from Absolute Convergence?
In absolute convergence, both the series ∑an and the series of its absolute values ∑∣an∣ converge. In conditional convergence, only ∑an converges, while ∑∣an∣ diverges.
Can you give an example of a Conditionally Convergent series?
A classic example is the alternating harmonic series: \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.
This series converges to ln(2), but the series of absolute value diverges.
What is the Riemann Series Theorem?
The Riemann Series Theorem states that a conditionally convergent series can be rearranged to converge to any real number or even to diverge.
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