Take this sentence:
Tom fancies Sarah.
"fancies" here is a finite verb -- it has a subject and it agrees with the subject: that means it changes, depending on what the subject is. "I fancy", but "Tom fancies".
I am working.
Here, "working" is a participle -- in fact, it's a present participle.
They have gone.
Here, "gone" is a past participle.
An infinitive is not finite and it's not a participle. Look at this famous quote:
To be or not to be...
"to be" is an infinitive. It doesn't have a subject that it could agree with.
Often an infinitive is easy enough to recognise, as it has the particle "to". Find the infinitives in this equally famous quote:
These are the voyages of the Starship Enterprise; its five-year mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man has gone before.
Note that "to boldly go" is an example of a split infinitive: the word "boldly" goes between "to" and "go", splitting it. Some purists consider this to be bad grammar.
An infinitive can sometimes occur without "to", as here:
Who does he think he is?
"Does" is a finite verb; it agrees with "he". It doesn't mean anything though, it just helps out with the grammar. It is followed by an infinitive (in this case "think") -- we say that the verb "to do" takes the infinitive without "to". I sometimes call this the "basic form" of the verb, just to avoid confusion.
English can be tricky in this respect, because often finite verbs happen to look exactly like their infinitives. For example:
Infinitive: to play
I play, you play, he plays, she plays, it plays, we play, they play
I play the piano. ("play" is finite and agrees with "I")
I can play the piano. ("play" is an infinitive in this sentence; the finite verb is "can")
In English, some verbs don't have infinitives. They are the modal verbs can, will, shall, ought and must.
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