Write one sentence that uses a participial phrase and an infinitive.

Ans. (one of many possible)

Badgering the old man about his cologne for weeks, Lenny decided to spray him down with Pine Sol.

Badgering the old man about his cologne for weeks, Lenny decided to spray him down with Pine Sol.

Here there is a simple present participial phrase at the beginning of the sentence.

Badgering the old man about his cologne for weeks

It is not dangling because Lenny is in fact doing this action.

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The infinitive (to spray) comes after the main verb in the main independent clause (in this case, "decided")

[Score 5 pts for a sentence that employs effective usage of these two parts of speech]