Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
1. **Determine \( f(0) \):**
   Given \( f \) is infinitely differentiable and \( f\left(\frac{1}{n}\right) = \frac{n^2}{n^2 + 1} \) for \( n = 1, 2, 3, \ldots \), we need to find \( f(0) \). Notice that as \( n \to \infty \), \( \frac{1}{n} \to 0 \). Therefore, we can use the definition of continuity:
   \[
   f(0) = \lim_{n \to \infty} f\left(\frac{1}{n}\right) = \lim_{n \to \infty} \frac{n^2}{n^2 + 1} = 1.
   \]

2. **Compute \( f'(0) \):**
   The first derivative at \( x = 0 \) is given by:
   \[
   f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac{f(x) - 1}{x}.
   \]
   Using the sequence \( x_n = \frac{1}{n} \), we have:
   \[
   f'\left(0\right) = \lim_{n \to \infty} \frac{f\left(\frac{1}{n}\right) - 1}{\frac{1}{n}} = \lim_{n \to \infty} \frac{\frac{n^2}{n^2 + 1} - 1}{\frac{1}{n}} = \lim_{n \to \infty} \frac{\frac{n^2 - (n^2 + 1)}{n^2 + 1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{\frac{-1}{n^2 + 1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{-n}{n^2 + 1} = 0.
   \]

3. **Compute \( f''(0) \):**
   The second derivative at \( x = 0 \) is given by:
   \[
   f''(0) = \lim_{x \to 0} \frac{f'(x) - f'(0)}{x - 0}.
   \]
   Since \( f'(0) = 0 \), we have:
   \[
   f''(0) = \lim_{x \to 0} \frac{f'(x)}{x}.
   \]
   Using the sequence \( x_n = \frac{1}{n} \), we need to find \( f'(x_n) \). Since \( f(x) \approx 1 - x^2 \) for small \( x \) (from the Taylor expansion around 0), we have:
   \[
   f'(x) \approx -2x.
   \]
   Therefore,
   \[
   f''(0) = \lim_{n \to \infty} \frac{-2 \cdot \frac{1}{n}}{\frac{1}{n}} = -2.
   \]

4. **Compute higher-order derivatives:**
   For \( k \geq 3 \), we need to consider the higher-order terms in the Taylor expansion of \( f(x) \). Since \( f(x) \approx 1 - x^2 \) for small \( x \), the higher-order derivatives at \( x = 0 \) will be zero. Thus, for \( k \geq 3 \):
   \[
   f^{(k)}(0) = 0.
   \]

The final answer is \( \boxed{ f(0) = 1 } \), \( f'(0) = 0 \), \( f''(0) = -2 \), and \( f^{(k)}(0) = 0 \) for \( k \geq 3 \).