1992 Putnam A4

Let $f$ be an infinitely differentiable real-valued function

defined on the real numbers. If

$$

f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,

$$

compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.

令 $f$ 为无限可微实值函数

定义在实数上。如果

$$

f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,

$$

计算导数 $f^{(k)}(0), k = 1, 2, 3, \dots$ 的值。