2016 Putnam B1

Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n \geq 0$,

$$

x_{n+1} = \ln(e^{x_n} - x_n)

$$

(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series

$$

x_0 + x_1 + x_2 + \cdots

$$

converges and find its sum.

设 $x_0,x_1,x_2,\dots$ 为满足 $x_0=1$ 的序列，对于 $n \geq 0$，

$$

x_{n+1} = \ln(e^{x_n} - x_n)

$$

(像往常一样，函数 $\ln$ 是自然对数)。证明无穷级数

$$

x_0 + x_1 + x_2 + \cdots

$$

收敛并求其和。