Changing the line type along a path is not so easy, so I guess it's not worth to implement it if this is the only place to apply. A quick fix can be applied by the following:
\begin{tikzpicture}[every node/.style={draw,circle}]
\draw[style=help lines] (-1cm,0cm) grid[step=1cm] (5cm,5cm);% remove later
\node (P0) at (0, 4cm) {$P_0$};
\node (P1) at (0, 2.5cm) {$P_1$};
\node (Pn) at (0, 1cm) {$P_n$};
\node (Q) at (3.5cm, 2.5cm) {$Q$};
\coordinate (Qf) at ([xshift=-0.5cm]Q.west); % we collect the edges in front of Q
\draw (P0) .. controls (2,4) and (1,2.5) .. (Qf) -- (Q); % (Q) is for a better line join
\draw (P1) -- (Qf);
\draw[->] (Qf) -- (Q); % the arrow
\draw (Qf) arc (90:150:1cm) coordinate (temp1); %We stop and change line type
\draw[dashed] (temp1) arc (-45:-80:2cm and 3.5cm) coordinate (temp2); % Again
\draw (temp2) -- (Pn);
\end{tikzpicture}
Now I see that there is a space between Pn and the rest, but that can be fixed by changing the arc specs.
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