Solution
= Solution
The probability generating <function> is
$$
G_X(t)=\mathbb E[t^X]=\sum_{k=0}^\infty p_kt^k,
\qquad 0\leq t\leq1.
$$
Termwise <differentiation> gives
$$
G_X'(t)=\sum_{k\geq1}kp_kt^{k-1},
\qquad
G_X''(t)=\sum_{k\geq2}k(k-1)p_kt^{k-2}.
$$
The coefficients are nonnegative, and $\mathbb P(X\geq2)>0$, so both <derivatives> are positive and nondecreasing on $(0,1]$.