Suppose
$$f(x) = 1 + x + x^2/2^2 + x^3/3^2 + … + x^k/k^2 + \cdots $$
What is the value of $f(0)$? $1$
What is the value of $f'(0)$ ? $1$
What is the value of $f''(0)$ ? $1/2$
What is the value of $f'''(0)$ ? $1/3$
However my answer for $f'''(0)$ is wrong. I don't understand why
$$\frac{1}{3^2}=\frac{1}{3!}f'''(0)$$
Multiply both sides by $3!$.