2024 Putnam B6

For a real number $a$, let $F_a(x) = \sum_{n \geq 1} n^a e^{2n} x^{n^2}$ for $0 \leq x < 1$.

Find a real number $c$ such that

$$

\begin{aligned}

& \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = 0 \qquad \text{for all $a < c$, and} \\

& \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = \infty \qquad \text{for all $a > c$.}

\end{aligned}

$$

对于实数 $a$，令 $F_a(x) = \sum_{n \geq 1} n^a e^{2n} x^{n^2}$ 对于 $0 \leq x < 1$。

找到一个实数 $c$ 使得

$$

\开始{对齐}

& \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = 0 \qquad \text{对于所有 $a < c$，并且} \\

& \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = \infty \qquad \text{对于所有 $a > c$.}

\结束{对齐}

$$