📝 题目
例 11 利用级数收敛性, 证明序列
$$ {x}_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} - \ln n $$
当 $\displaystyle{n \rightarrow \infty}$ 时极限存在.
💡 答案与解析
证 令 ${a}_{1} = {x}_{1},{a}_{n} = {x}_{n} - {x}_{n - 1}\left( {n = 2,3,\cdots }\right)$ ,则级数 $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}}$ 的部分和为 ${x}_{n}$ ,所以证 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n}}$ 存在归结为证级数收敛. 因
$$ {a}_{n} = \frac{1}{n} + \ln \left( {1 - \frac{1}{n}}\right) = \frac{1}{n} + \left\lbrack {-\frac{1}{n} - \frac{1}{2{n}^{2}} + o\left( \frac{1}{{n}^{2}}\right) }\right\rbrack $$
$$ = - \frac{1}{2{n}^{2}} + o\left( \frac{1}{{n}^{2}}\right) , $$
由于 ${a}_{n} \sim - \frac{1}{2{n}^{2}}$ ,推出级数 $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}}$ 收敛,也就是 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = c}$ 存在, $c$ 称为欧拉常数, $c = {0.577216}\cdots$ . 若记 ${x}_{n} - c = {r}_{n}$ ,则
$$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = c + \ln n + {r}_{n},\;\mathop{\lim }\limits_{{n \rightarrow \infty }}{r}_{n} = 0. $$