Convergence of an Alternating Series: The Alternating Harmonic Series Explained

The concept of an alternating series is quite intriguing and is a fundamental topic in the study of calculus. One of the most well-known and instructive examples is the alternating harmonic series. This article will delve into the properties and convergence of the alternating harmonic series. We will use the theorem of Leibniz to analyze its behavior and provide a comprehensive understanding of why and how it converges.

The Alternating Harmonic Series

Consider the series:

$$ S_n 1 - frac{1}{2} frac{1}{3} - frac{1}{4} cdots (-1)^{n-1} frac{1}{n} - (-1)^n frac{1}{n 1} $$

For this series, let's define the terms and see why it starts with the first term as (u_1 1).

Understanding the Terms

The terms of the series can be written as:

$$ u_n (-1)^{n-1} frac{1}{n} $$

It is clear that the first term is defined as:

$$ u_1 1 $$

Hence, the series starts with (u_1 1), as the original term is not defined for (n0).

Properties of Successive Differences

Let's now examine the differences between successive terms:

$$ d_n u_{n-1} - u_n frac{1}{n-1} - frac{1}{n} frac{1}{(n-1)n} $$

The differences (d_n) are all positive and form a decreasing sequence. This is evident from the expression for (d_n).

Convergence Analysis Using Leibniz's Theorem

To analyze the convergence of the alternating series, we can apply the Leibniz Theorem. This theorem states that an alternating series of the form:

$$ u_1 - u_2 u_3 - u_4 cdots (-1)^{n-1}u_n (-1)^n u_{n 1} $$

where (u_n) is a positive sequence and (u_n downarrow 0) (i.e., (u_n) monotonically decreases to 0), then the series converges.

In the case of the alternating harmonic series, we have:

$$ u_n frac{1}{n} $$

It is straightforward to show that:

$$ u_n text{ is positive and monotonically decreasing to } 0. $$

Therefore, by Leibniz's Theorem, the alternating harmonic series converges. This conclusion is supported by the textbook G.M. Fihtenholjc 1964 (page 278) and several other calculus references.

Non-Absolutely Convergence

While the alternating harmonic series is convergent, it is important to note that it is not absolutely convergent. This can be shown by considering the harmonic series:

$$ sum_{n1}^{infty} frac{1}{n} $$

This series is known to diverge to infinity, as the sum of the reciprocals of the natural numbers grows without bound. Therefore, the alternating harmonic series is conditionally convergent.

Further Reading and Resources

For a deeper understanding of the subject, you may consult:

"Course of Differential and Integral Calculus" by G.M. Fihtenholjc, Volume II (in Romanian, translated from Russian), Technical Publishing House Bucharest, 1964.

The properties and behavior of the alternating harmonic series are a classic example in calculus and offer valuable insights into the nature of series convergence.