2011 Putnam A2

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive

real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for

$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove that

$$

S = \sum_{n=1}^\infty \frac{1}{a_1...a_n}

$$

converges, and evaluate $S$.

设 $a_1,a_2,\dots$ 和 $b_1,b_2,\dots$ 为正数序列

实数，使得 $a_1 = b_1 = 1$ 且 $b_n = b_{n-1} a_n - 2$

$n=2,3,\点$。假设序列 $(b_j)$ 是有界的。证明

$$

S = \sum_{n=1}^\infty \frac{1}{a_1...a_n}

$$

收敛，并评估$S$。