I think it was not only the teachings of Gandalf that led Frodo to extend compassion to Gollum, but also the fact that Frodo knew that, through the influence of the Ring, he could very well have become Gollum.
No one but Frodo could understand the overwhelming curse that the power of the One Ring could lay on its bearer, particularly as he got closer and closer to Mordor. I'm sure Frodo would have agreed completely with the old saying "There, but for the grace of God, go I."
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