the cumulant generating function
The mgf is a moment factory, but moments combine awkwardly under addition — the moments of a sum involve cross-terms and are fiddly. There is a small twist on the mgf that fixes this and makes adding independent variables feel almost effortless: take the logarithm. That single move converts the mgf's multiplication rule into an addition rule, which is exactly what you want when you keep stacking up independent pieces.
The cumulant generating function is just K_X(t) = log(M_X(t)) = log E[e^(tX)]. Why bother with a logarithm? Because for independent X and Y the mgf multiplies, M_{X+Y}(t) = M_X(t) M_Y(t), and the log of a product is a sum, so K_{X+Y}(t) = K_X(t) + K_Y(t). The cgf of a sum is the plain sum of the cgfs. Expanding K_X(t) as a power series, K_X(t) = kappa_1 t + kappa_2 t^2/2! + kappa_3 t^3/3! + ..., the coefficients kappa_n are the cumulants, and because the whole cgf adds, every cumulant adds too. The first two are friendly faces in disguise: kappa_1 = E[X] (the mean) and kappa_2 = Var(X) (the variance).
This additivity is why the cgf is the natural tool for studying sums, and it is the cleanest route into the central limit theorem: when you add n independent identical copies, the cgf simply multiplies by n, and after the standard rescaling the higher cumulants shrink away leaving only the mean and variance — the normal's signature, since the normal is the unique distribution whose cumulants past the second are all zero. The same honest caveat as the mgf applies: the cgf requires the mgf to exist near 0, so for heavy-tailed variables it is unavailable.
For X ~ Normal(mu, sigma^2), M_X(t) = e^(mu t + sigma^2 t^2/2), so K_X(t) = log M_X(t) = mu t + sigma^2 t^2/2. The expansion stops dead after the t^2 term: kappa_1 = mu, kappa_2 = sigma^2, and every higher cumulant is exactly 0 — a clean signature that the third-and-beyond cumulants vanishing means 'normal'.
K_X(t) = log M_X(t): its power-series coefficients are the cumulants, and it ADDS for independent sums.
The cgf inherits the mgf's limitation: it needs the mgf to be finite near 0. Where the mgf does not exist, neither does the cgf.