Chapter 11.3, Problem 44E

Use the following steps to show that the sequence

$t_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\ln n$

has a limit. (The value of the limit is denoted by γ and is called Euler’s constant.)

1. (a) Draw a picture like Figure 6 with f(x) = 1/x and interpret t_n as an area [or use (5)] to show that t_n > 0 for all n.
2. (b) Interpret

$t_n-t_{n+1}=[\ln (n+1)-\ln n]-\frac{1}{n+1}$

as a difference of areas to show that t_n − t_n+1 > 0. Therefore {t_n} is a decreasing sequence.

1. (c) Use the Monotonic Sequence Theorem to show that {t_n} is convergent.