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evaluate sum of partial products

Source: 2019 South Korea USMC P5

August 13, 2020
Sequencesseriesexponentiallogarithmscollege contests

Problem Statement

A sequence {an}n≥1\{a_n\}_{n\geq 1} is defined by a recurrence relation a_1 = 1,  a_{n+1} = \log \frac{e^{a_n}-1}{a_n} And a sequence {bn}n≥1\{b_n\}_{n\geq 1} is defined as bn=∏i=1naib_n = \prod\limits_{i=1}^n a_i. Evaluate an infinite series ∑n=1∞bn\sum\limits_{n=1}^\infty b_n.
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