Exploring Convergent Series That Aren’t Absolutely Convergent: The Alternating Harmonic Series
Understanding the behavior of series in mathematics is a fundamental aspect of analysis. A common question that arises is whether a series that converges must also converge absolutely. The answer is generally no, and there are several well-known examples of this phenomenon. This article delves into one such example: the Alternating Harmonic Series and provides a detailed explanation of its properties.
Introduction to the Alternating Harmonic Series
The Alternating Harmonic Series is a classic example in mathematical analysis. It is defined as follows:
$$ sum_{n1}^{infty} frac{(-1)^{n 1}}{n} $$
which translates to the series: 1 - 1/2 1/3 - 1/4 1/5 - ….
Convergence of the Alternating Harmonic Series
The Alternating Harmonic Series is a good starting point because it converges. This can be demonstrated using the Alternating Series Test, also known as the Leibniz Test, which applies to series of the form: $$ sum_{n1}^{infty} (-1)^{n 1} a_n $$, where ( a_n ) is a sequence of positive terms that are decreasing and approach zero.
For the Alternating Harmonic Series, ( a_n frac{1}{n} ). Clearly, ( a_n ) is a decreasing sequence that converges to zero as ( n ) tends to infinity. Hence, the series converges by the Alternating Series Test.
The Divergence of the Harmonic Series
Contrast the Alternating Harmonic Series with the Harmonic Series, which is the sum of the positive terms: $$ sum_{n1}^{infty} frac{1}{n} $$, or 1 1/2 1/3 1/4 ….
The Harmonic Series is a classic case of a series that diverges. There are various ways to show this, including the Divergence Test and comparison tests. One simple method is to group the terms in a way that the series diverges:
$$ left(1right) left(frac{1}{2}right) left(frac{1}{3} frac{1}{4}right) left(frac{1}{5} frac{1}{6} frac{1}{7} frac{1}{8}right) cdots $$
Each group in brackets sums to more than 1/2, making the sum of the series infinite.
Convergence vs. Absolute Convergence
A series is said to be absolutely convergent if the series of absolute values of its terms converges. To determine if the Alternating Harmonic Series is absolutely convergent, we need to evaluate the series:
$$ sum_{n1}^{infty} left| frac{(-1)^{n 1}}{n} right| sum_{n1}^{infty} frac{1}{n} $$
This is just the Harmonic Series, which diverges. Therefore, the Alternating Harmonic Series is not absolutely convergent.
Implications and Further Reading
The fact that a series can converge without being absolutely convergent has significant implications in analysis. It means that the order of summation matters for the sum of the series. For example, the Riemann Series Theorem states that if a series is conditionally convergent, then the series can be rearranged to converge to any real number, or even diverge.
Further reading on series convergence and divergence can be found in standard texts on mathematical analysis, such as "Principles of Mathematical Analysis" by Walter Rudin and "Introduction to Real Analysis" by Robert G. Bartle and Donald C. Sherbert.
Keywords: convergent series, absolutely convergent, alternating harmonic series