Question

Let \(S_{n}=\sum_{k=1}^{n} \frac{1}{k}=1+\frac{1}{2}+\cdots+\frac{1}{n} .\) (a) Show that \(\ln (n+1) \leq S_{n} \leq 1+\ln n\) (b) Show that the sequence \(\left\\{a_{n}\right\\}=\left\\{S_{n}-\ln n\right\\}\) is bounded. (c) Show that the sequence \(\left\\{a_{n}\right\\}\) is decreasing. (d) Show that \(a_{n}\) converges to a limit \(\gamma\) (called Euler's constant). (e) Approximate \(\gamma\) using \(a_{100}\).

Short answer

a) Prove an inequality using integral approximation: \(\ln (n+1) \leq S_{n} \leq 1+\ln n\). b) Show that \(a_{n}=S_{n}-\ln n\) is bounded above and below. c) Demonstrate that the sequence is decreasing by showing that \(a_{n+1}\leq a_{n}\). d) By the Monotone Convergence Theorem, the sequence \(a_{n}\) is convergent to a limit \(\gamma\) (Euler's constant). e) By taking \(n=100\) in the expression for \(a_n\), you can estimate the value of Euler's constant.

Step-by-step solution

Part A: Show Inequality

To prove \(\ln (n+1) \leq S_{n} \leq 1+\ln n\), first notice that \(1/x\) is decreasing and by integral approximation, \[\int_{1}^{n} \frac{1}{x} d x \leq S_{n} \leq 1+\int_{1}^{n} \frac{1}{x} d x\]. Computing these integrals gives: \(\ln n \leq S_{n} \leq 1+\ln (n-1)\), which is equivalent to what we want to prove.

Part B: Show Boundedness

To show that the sequence \(a_{n}=S_{n}-\ln(n)\) is bounded, find its limits as n approaches infinity. Observe from the previous part that a lower bound can be 0 and an upper bound \(1\), which means that the sequence is bounded.

Part C: Show Decreasing Nature

Note that \(a_{n+1}-a_{n}=S_{n+1}-\ln (n+1)-(S_{n}-\ln n)=\frac{1}{n+1}-\ln \left(\frac{n+1}{n}\right)=-\ln \left(1-\frac{1}{n+1}\right)\leq 0\) (since -ln(1-x) ≤ 0 for 0 < x < 1). Hence, \( a_{n+1}\leq a_{n}\) for all n, meaning the sequence is decreasing.

Part D: Show Convergence

As we proved in previous parts, \(a_{n}=S_{n}-\ln(n)\) is a sequence that is decreasing and bounded below by 0. By Monotone Convergence Theorem, \(a_{n}\) is then convergent. Its limit as n approaches infinity is denoted by Euler's constant, \(\gamma\).

Part E: Approximate Euler's Constant

To approximate Euler's constant, we substitute \(n=100\) into the sequence. Compute \(S_{100}- \ln(100)\) to get an approximate value for Euler's constant.

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two main branches: differential calculus, which concerns itself with the concept of a derivative, and integral calculus, which deals with the concept of an integral. In the context of Euler's constant and the exercise at hand, integral calculus plays a key role. Integral calculus, among other things, helps us understand the area under a curve, which is closely related to summing up the values of sequences.

For example, to prove that \(\ln (n+1) \leq S_{n} \leq 1+\ln n\), it's crucial to consider the integral \(\int_{1}^{n} \frac{1}{x} dx\), which represents the area under the curve of the function \(f(x) = \frac{1}{x}\) from 1 to \(n\). This area provides an approximation for the sum \(S_{n}\), giving us a way to establish that the inequality holds. Using integrals to approximate sums is an essential technique in calculus for analyzing complex series and sequences. This is a foundational concept in understanding the behavior of the harmonic series involved in the definition of Euler's constant.

Sequences and Series

Sequences and series are fundamental concepts in mathematics, particularly within the study of calculus. A sequence is an ordered list of numbers that often follow a specific rule or pattern, while a series is the sum of the terms of a sequence. The behavior of sequences and their associated series can shed light on many intricate mathematical phenomena.

In this exercise, the harmonic series, denoted by \(S_{n} = \sum_{k=1}^{n} \frac{1}{k}\), is a central focus. Demonstrating that this series has certain bounds and understanding its approach to a limit as \(n\) increases aids in our comprehension of convergence and divergence — core topics in the study of sequences and series. Part (b) of the exercise, which establishes that the modified sequence \(\{a_{n}\} = \{S_{n} - \ln n\}\) is bounded, and part (c), which shows this sequence is also decreasing, are both tied to the fundamental properties of sequences and series. These properties must be grasped to understand the convergence of sequences, which leads to the concept of limits and ultimately to the value known as Euler's constant.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm maps multiplication into addition, which becomes particularly useful when dealing with growth processes or rates of decay in various scientific and mathematical contexts.

In the context of Euler's constant, the natural logarithm intersects with the concept of the harmonic series. For instance, the inequality \(\ln (n+1) \leq S_{n} \leq 1+\ln n\) in part (a) intertwines the natural logarithm with the harmonic series. Furthermore, the sequence \(\{a_{n}\}\) involves the natural logarithm as part of its expression. As part (c) suggests, analyzing the decreasing nature of the sequence \(\{a_{n}\}\) requires understanding the behavior of the natural logarithm function. By comprehending how the natural logarithm behaves in this sequence, students can unravel the steps that lead to proving the convergence toward Euler's constant, which is pivotal in advancing their grasp of the interplay between sequences, series, and logarithmic functions.