2016 Putnam Problem B1

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 + \dots converges and find its sum.

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