= Solution
Because $s>1$, the <function> $t\mapsto s^t$ is increasing and nonnegative. Thus $\{X\geq x\}=\{s^X\geq s^x\}$. Applying <Markov's inequality> to the nonnegative <random variable> $s^X$, whose <expected value> is finite by assumption, yields
$$
\boxed{\mathbb P(X\geq x)\leq\frac{\mathbb E[s^X]}{s^x}=s^{-x}G_X(s).}
$$
Under the usual <probability generating function> convention $X$ takes nonnegative <integer> values. The argument itself works for any real <random variable> if $G_X(s)$ is understood as the finite <expected value> $\mathbb E[s^X]$.
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