Question

The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise. In certain applications of probability, such as the so-called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate estimate of the number \(H_{k}=\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{k}\right) .\) Recall that \(T_{k}=H_{k}-\ln k\) is decreasing. Compute \(T=\lim _{k \rightarrow \infty} T_{k}\) to four decimal places. (Hint: \(\left.\frac{1}{k+1}<\int_{k}^{k+1} \frac{1}{x} d x .\right)\)

Short answer

The limit is approximately 0.5772.

Step-by-step solution

Understand the Given Expression

The expression for partial sums of the harmonic series is given as \( H_k = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} \). This series needs to be analyzed to compute the limit of the difference between \( H_k \) and the natural logarithm \( \ln k \), denoted as \( T_k = H_k - \ln k \). We need to find \( \lim_{k \rightarrow \infty} T_k \).

Understand the Decreasing Property of \( T_k \)

Given that \( T_k \) is decreasing, this implies that the difference between \( H_k \) and \( \ln k \) gets smaller as \( k \) increases. Thus, the limit of \( T_k \) as \( k \) approaches infinity should converge to a fixed value.

Use the Integral Test

By applying the hint, which suggested that \( \frac{1}{k+1}<\int_{k}^{k+1} \frac{1}{x} d x \) holds true, we use the fact that: $$ H_k \approx \ln k + \gamma $$ where \( \gamma \) is the Euler-Mascheroni constant. Therefore, \( T_k = H_k - \ln k \) converges to this constant \( \gamma \).

Compute \( \gamma \) and the Limit

The value of the Euler-Mascheroni constant \( \gamma \) is approximated to be 0.5772. Hence, we conclude that the limit \( T = \lim_{k \rightarrow \infty} T_k = \gamma \), and is approximately 0.5772.

Key concepts

Watterson Estimator

The Watterson estimator is a valuable tool in population genetics, allowing scientists to predict the mutation rate within a population. Named after geneticist Dr. Watterson, this estimator is closely linked to the concept of the harmonic series. When we study mutation rates, understanding how these partial sums behave is crucial. In essence, the Watterson estimator provides insight into the genetic variation by using the number of segregating sites in a sample of DNA sequences. This is often modeled through the harmonic sum, represented by \( H_k \), which adds inverse terms up to \( \frac{1}{k} \).
What makes this tool particularly powerful is its ability to adjust for the effective population size when predicting mutation rates, offering a historical glimpse into genetic variations. So, when calculating \( H_k \), the estimator uses the cumulative partial sums to achieve a more stable prediction. This method underscores the importance of understanding harmonic series in broader mathematical and scientific applications.

Partial Sums

Partial sums are a mathematical way to add up a sequence of numbers up to a certain point and play an essential role in understanding series, such as the harmonic series. For instance, the partial sum of the harmonic series \( H_k = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} \) is used frequently in various mathematical fields.
This concept can be visualized by stacking fractions that steadily decrease in value. While each individual fraction is relatively small, collectively, they provide insightful information about the behavior of series in the limit.
Partial sums indicate the cumulative total of the first \( k \) terms of a series and are especially important in determining convergence or divergence of a series. As partial sums increase, they offer a snapshot of the entire series, as seen in many practical applications. This is where the relationship to logarithmic functions and constants like Euler-Mascheroni come into play. They help mathematicians approximate and understand the infinite behaviors of these sums.

Euler-Mascheroni Constant

The Euler-Mascheroni constant, denoted as \( \gamma \), is a mesmerizing mathematical constant often appearing in number theory. It forms a bridge between harmonic series and logarithmic functions.
This constant is essentially the limit of the difference between the harmonic series and the natural logarithm as \( k \) approaches infinity.
The Euler-Mascheroni constant is defined as:

• \( \gamma = \lim_{n \to \infty} \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \ln(n) \right) \)

Known to be approximately 0.5772, \( \gamma \) embodies the delicacy of mathematical limits. Its presence in various forms throughout mathematics highlights its significance and usefulness in mathematical analysis.
Understanding the Euler-Mascheroni constant enables one to appreciate how we transition from discrete harmonics to continuous logarithms smoothly. It stands as a testament to how interconnected different areas of mathematics can be. Find it applied across topics ranging from calculus to complex analysis, showcasing its widespread influence. This constant aids in the approximate calculations and provides a deeper understanding of numerical behaviors. It serves as an ongoing reminder of how constants connect mathematical theories.