Problem 3 - Complex Analysis

Question

Show that the sequence \(\left(\gamma_{n}\right)\) defined by the expression $$ \gamma_{n}:=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n $$ is (strictly) decreasing, and bounded from below by 0 . Hence the following limit exists: $$ \gamma:=\lim _{n \rightarrow \infty} \gamma_{n} \approx 0,577215664901532860606512090082402431042159 \ldots $$

Step-by-Step Solution

Answer

The sequence \( \gamma_n \) is decreasing and bounded below by 0, so the limit \( \gamma_n \) exists.

Step 1: Understand the sequence

The sequence \( \gamma_n \) is defined by the sum \( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) minus \( \log n \). This can be expressed as \( \gamma_n = H_n - \log n \), where \( H_n \) is the \( n \)-th harmonic number.

Step 2: Analyze the derivative

To check if \( \gamma_n \) is decreasing, consider the difference \( \gamma_{n+1} - \gamma_{n} \). Calculate it as getting \( H_{n+1} - \log(n+1) - \gamma_n \). This leads to finding \( (H_{n+1} - \log(n + 1)) - (H_n - \log n) \).

Step 3: Simplify the difference

Rewrite the difference: \( H_{n+1} - H_n = \frac{1}{n+1} \). The difference becomes \( \frac{1}{n+1} - (\log(n+1) - \log n) \), which is equal to \( \frac{1}{n+1} - \log \left(1 + \frac{1}{n}\right) \).

Step 4: Establish decreasing condition with inequality

For the sequence to be decreasing, we need \( \gamma_{n+1} - \gamma_{n} \leq 0 \). Notice that \( \log \left(1 + \frac{1}{n}\right) \approx \frac{1}{n} - \frac{1}{2n^2} \ldots \) for large \( n \). This approximation shows that \( \log \left(1 + \frac{1}{n}\right) > \frac{1}{n+1} \), hence \( \gamma_{n+1} < \gamma_n \), confirming the sequence is decreasing.

Step 5: Check boundedness

Since for large \( n \), \( H_n \approx \log n + \gamma \), \( \gamma_n = H_n - \log n \approx \gamma \). \( \gamma_n \) is positive and tends towards a constant value, so it is bounded below by 0.

Step 6: Establish the limit

Given \( \gamma_n \) is bounded and decreasing, it converges to a limit. Let's find \( \gamma = \lim_{n \to \infty} \gamma_n \approx 0.577215\ldots \). This limit is the Euler-Mascheroni constant.

Key Concepts

Decreasing Sequences

A decreasing sequence is a sequence where each term is less than or equal to the term before it. For the sequence \(\gamma_n\), it is defined by taking the sum of the harmonic numbers and subtracting the logarithmic function of \(n\). To determine if this sequence is strictly decreasing, we look at the difference between consecutive terms. This is expressed as \(\gamma_{n+1} - \gamma_n\). If this difference is negative, the sequence is decreasing.

In the case of \(\gamma_n\), the analysis shows that \((H_{n+1} - H_n) - (\log(n+1) - \log n)\) simplifies to \(-\log(1+\frac{1}{n}) + \frac{1}{n+1}\). Using approximations for large \(n\), specifically that \(\log(1+\frac{1}{n}) > \frac{1}{n+1}\), we can see \(\gamma_{n+1} < \gamma_n\). Thus, the sequence is confirmed to be decreasing.

This method hinges on the property of logarithms and the behavior of harmonic numbers as \(n\) increases, emphasizing the sequence's nature over an infinite set.

Logarithmic Function

A logarithmic function \(\log x\) is a mathematical function which is the inverse of the exponential function. It helps to determine the power to which a base number must be raised to obtain a given number. In our sequence \(\gamma_n\), the logarithmic function is used to offset the growth of the harmonic numbers.

The sequence incorporates the natural logarithm \(\log n\) which acts to slow down the increasing nature of the sum of reciprocals as it becomes larger. The difference between consecutive logarithmic values \(\log(n+1) - \log n\) is used to analyze their influence on the sequence. This difference simplifies to \(\log(1 + \frac{1}{n})\), and crucially, it helps us establish the conditions for which \(\gamma_n\) is decreasing.

Logarithms decrease slowly for large values of \(n\), which is significant because while the harmonic series grows, its growth is subtracted by this logarithm, causing the sequence to approach a limit.

Euler-Mascheroni Constant

The Euler-Mascheroni constant, often symbolized as \(\gamma\), is a mathematical constant that appears in various problems of number theory. This constant has an approximate value of 0.577215. It is defined as the limiting difference between the harmonic series and the natural logarithm.

In the context of our sequence \(\gamma_n\), as \(n\) tends to infinity, \(\gamma_n\) converges toward this constant. This convergence arises because as we subtract the slowly diverging logarithmic function from the harmonic sum, what's left approaches a stable value, which is \(\gamma\).

The Euler-Mascheroni constant is significant not only in this specific sequence but also in fields like calculus and analytic number theory. It reflects the inherent balance between the growth rates of sums of reciprocals and logarithms, providing a snapshot of a crucial asymptotic behavior in mathematics.