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Putnam 2016 B1

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Putnam 2016 B1

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

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

Show that the infinite series $\sum \limits_{n=0}^{\infty} x_n$ converges and evaluate its value.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$