Chapter 10 Conceptual Project:

Working in Harmony

In this project, we are going to expand on our earlier work with the harmonic series. In the process, we will meet a famous constant called Euler's constant, also known as the Euler-Mascheroni constant. (This number is not to be confused with $e\approx 2.71828$, the natural base, which is also known as Euler's number.)

1. As in Example 6 of Section 10.2, we let $s_n$ stand for the n^th partial sum of the harmonic series; that is,

   $s_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}$.

   (The partial sum $s_n$ is also called the n^th harmonic number.) For each $n\ge 1$, we define

   $d_n=s_n-lnn$.

   Prove that $d_n>0$ for any positive integer n. (Hint: Refer to the illustration provided for Exercise 65 of Section 10.2, and start by comparing $s_n$ with ${\displaystyle {\int }_1^{n+1}\left({\displaystyle \frac{1}{x}}\right)\mathit{d}x}$.)

2. Prove that $\left\{d_n\right\}$ is a decreasing sequence. (Hint: Referring again to the figure from Exercise 65 of Section 10.2, fix an n and identify a region whose area is $d_n-d_{n+1}$ .)

3. Use an appropriate theorem from the text to show that the sequence $\left\{d_n\right\}$ is convergent. Letting $\gamma =\underset{n\to \infty }{\lim }{d}_n$, this limit is called Euler's constant. It is important in many applications throughout various areas of mathematics, and like other famous constants (including π and e) can be approximated with great precision using modern computing power. Surprisingly, however, it is not yet known whether γ is rational or irrational!

4. Use the convergence of $\left\{d_n\right\}$ to prove that the sequence $a_n={\displaystyle \sum _{i=n}^{2n}{\displaystyle \frac{1}{i}}}$ converges and find its limit.

5. Use a computer algebra system to approximate γ, accurate to the first $10$ decimal places.

6. Use the approximate value of γ found in Question 5 to estimate $s_n$, rounded to $5$ decimal places, for a. $n=\mathrm{10,000}$ and b. $n=\mathrm{2,000,000}$. Compare the latter estimate with the answer for Exercise 125b of the Chapter 10 Review.