Only the first term of some series counts

Let X,X1,X2,… be i.i.d. random variables, and set Sn=X1+⋯+Xn. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type ∑n≥1anP(|Sn|≥xbn) or ∑n≥1anP(Sn≥xbn) behave like their first term as x→∞.