设$r_{n}$是$\tan{x}=x$的第$n$个正根，则有
\[ r_{n}=R_{n}-\frac{1}{R_{n}}-\frac{2}{3}\frac{1}{R^3_{n}}-\frac{13}{15}\frac{1}{R_{n}^{5}}-\frac{146}{105}\frac{1}{R_{n}^{7}}-\frac{781}{315}\frac{1}{R_{n}^{9}}+o\left(\frac{1}{R^{10}_{n}} \right)\]
这里
\[ R_{n}=\left(n+\frac{1}{2}\right)\pi\]