Question

The series \(\sum_{n=1}^{\infty} \frac{1}{2 n}\) is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which \(n\) is odd. Let \(m>1\) be fixed. Show, more generally, that deleting all terms \(1 / n\) where \(n=m k\) for some integer \(k\) also results in a divergent series.

Short answer

Removing a finite fraction of terms from a divergent series leaves a divergent series.

Step-by-step solution

Understanding the Series Modification

We start with the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) and consider deleting all terms where \( n = mk \) for some integer \( k \) and fixed \( m > 1 \). This means making a new series by keeping only terms where \( n \) is not a multiple of \( m \). We aim to show that this new series diverges.

Defining the New Series

Consider the series \( \sum_{n=1, n ot\equiv 0 \pmod{m}}^{\infty} \frac{1}{n} \). This series includes all terms from the harmonic series except those divisible by \( m \).

Utilizing Comparison Test

By the nature of the harmonic series, which is known to diverge, any series formed by removing a finite fraction of terms from it should still diverge. The sum of the terms we remove, \( \sum_{k=1}^{\infty} \frac{1}{mk} \), is an arithmetic subseries of the harmonic series and diverges, but still has an infinite sum. Therefore, the remaining terms do not sum to a finite amount.

Concluding Divergence

Since removing an infinite number of terms corresponding to \( n=mk \) leaves a series where the remaining terms are not a negligible fraction of the harmonic series, it must also diverge according to the comparison test with a divergent series.

Key concepts

Comparison Test

The Comparison Test is a powerful tool for analyzing the convergence or divergence of an infinite series. When you want to determine if a series diverges or converges, you can compare it with another series whose behavior is already known. By comparing with a well-understood series, you gain insights into the original series.

Here's how the Comparison Test works:

• If you compare your series with a divergent series and your series is larger, your series diverges too.
• If your series is smaller than a convergent series, then it converges.

In the context of this exercise, we use the Comparison Test on a modified harmonic series. This series excludes terms making the fractions smaller but not enough to change the nature of the whole sequence. Since the harmonic series diverges, removing certain multiples will still result in a divergent series. The intuition here is essential: removing a finite fraction of an infinite, divergent series does not change its divergence behavior.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. It is expressed in the form \(\sum_{n=1}^{\infty} a_n \), where each term \(a_n\) is a part of the series. Understanding whether these infinite series converge (sum to a finite number) or diverge (goes to infinity or does not settle at a number) is fundamental in calculus.

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is one of the most famous examples of an infinite series. Despite its terms shrinking to zero, the series itself diverges. This result is crucial because it shows that not all series of positive, decreasing terms converge.

In this exercise, by removing specific terms from the harmonic series, we create a new infinite series. However, since we're only removing terms at regular intervals, the overall structure of the series remains similar to the harmonic series, maintaining its divergent nature.

Divergent Series

A divergent series is an infinite series that does not converge to a finite limit. When the sum of the terms in a series goes to infinity or simply fails to approach any specific finite value, the series is said to diverge.

The harmonic series, \(\sum_{n=1}^{\infty} \frac{1}{n} \), is a prime example of a divergent series—it grows without bound as more terms are added.

In the specific case of our exercise, deleting terms where \(n = mk\) (multiples of a fixed number \(m\)) still results in a divergent series. The key observation is that removing terms at regular, predictable intervals (like every other term, every third term, etc.) doesn't make the sum of the remaining terms finite. Instead, the series still preserves a "skeleton" of growth akin to that of the harmonic series, ensuring divergence. This is because, even after removal, an infinite number of terms still contribute to the series, and these terms are not small enough to sum to a finite value.