If we have an independent random variable X and a scalar constant alpha, show that $\kappa_n(\alpha X) = \alpha^n \kappa_n(X)$ holds by using the the Maclaurin series is given as $K_X(t) = \sum_{n=1}^\infty \kappa_n(X) \frac{t^n}{n!}$. The task specifies to let $\kappa_n(X) = K_X^{(n)}(0)$ be the $n$-th derivative of $K_X(t)$ evaluated at $t=0$.
The cumulant-generating function is defined as: $$K_X(t) = \log E[e^{tX}]$$
I started by just injecting $\alpha X$ into the formula for the Maclaurin series like so:
$$K_{\alpha X(t)} = \sum_{n=1}^\infty \kappa_n(\alpha X) \frac{t^n}{n!}$$
But now I'm stuck and given only the task description, I don't really know how to proceed, especially since cumulants are a new topic for me.
Any help or further hints would be appreciated!
$$K_{aX}(t)=\sum^\infty_{n=0}\frac{\kappa_n(aX)}{n!}t^n=\log(E[e^{taX}])=\log(E[e^{(ta)X}])=K_X(at)=\sum^\infty_{n=0}\frac{\kappa_n(X)}{n!}a^nt^n$$
Equating coefficients of the same order gives $\kappa_n(aX)=a^n\kappa_n(X)$