= Solution
Write $\tau=T-t>0$ and $g_n(s)=(K-s)_+^n$. The <European power put formula> is
$$
P_n(S,t)=e^{-r\tau}\mathbb E\!\left[(K-SY)_+^n\right],\qquad Y=e^{(r-\sigma^2/2)\tau+\sigma\sqrt\tau Z},\quad Z\sim N(0,1).
$$
Here $Y>0$ has a <lognormal distribution>. Since $g_n$ is decreasing and is a <convex function>, its discounted <expected value> is also decreasing and convex as a function of $S$. For $S>0$ it is smooth and strictly positive: the <normal distribution> assigns positive probability to $SY<K$ whatever the current positive spot price. Its endpoints are
$$
\boxed{P_n(0,t)=K^n e^{-r\tau},\qquad \lim_{S\to\infty}P_n(S,t)=0}.
$$
The <dominated convergence theorem> justifies the second <limit>, because $0\leq(K-SY)_+^n\leq K^n$. For a quantitative sketch, the <binomial theorem> and <truncated lognormal moment> give
$$
P_n(S,t)=e^{-r\tau}\sum_{j=0}^n(-1)^j\binom nj K^{n-j}S^j e^{jr\tau+j(j-1)\sigma^2\tau/2}\Phi(-d_j),\qquad d_j=\frac{\log(S/K)+[r+(j-\tfrac12)\sigma^2]\tau}{\sigma\sqrt\tau},
$$
where $\Phi$ is the <standard normal distribution function>. This <finite sum> is useful for drawing the curve; its positive <expectation> form makes its qualitative shape clearer.
\Image[/past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2023-power-put-prices.png]
{title=European power-put prices before expiry for powers one, two and three}
The solid curves use $K=1$, $r=0.05$, $\sigma=0.30$ and $\tau=1$; the dashed curves show their terminal <financial payoffs> for comparison. Each price remains positive at $S=K$ and tends smoothly towards zero beyond it. At small $S$, discounting gives a price below the undiscounted <financial payoff>; European exercise is permitted only at expiry.
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