Chapter 5.2, Problem 3SP

Euler’s Constant

We have shown that the harmonic series ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n}}$ diverges. Here we investigate the behavior of the partial S_ksums, as $k\to \infty$ . In particular, we show that they behave like the natural logarithm function by showing that there exists a constant such that

${\displaystyle \underset{n=1}{\overset{k}{\sum }}\frac{}{n}-\ln k\to \gamma \text{as}k\to \infty .}$

‘This constant $\gamma$ is known as Euler’s constant.

3. Now estimate how far $T_k$is from for a given integer k. Prove that for k $\ge$ 1. 0 < T_k — $\gamma$ $\le$k by using the following Steps.