1985 Putnam A3

Let $d$ be a real number. For each integer $m \geq 0$, define a

sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition

$$

\begin{aligned}

a_m(0) &= d/2^m, \\

a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0.

\end{aligned}

$$

Evaluate $\lim_{n \to \infty} a_n(n)$.

令 $d$ 为实数。对于每个整数 $m \geq 0$，定义一个

按条件序列 $\{a_m(j)\}$, $j=0,1,2,\dots$

$$

\开始{对齐}

a_m(0) &= d/2^m, \\

a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0。

\结束{对齐}

$$

评估 $\lim_{n \to \infty} a_n(n)$。