converge vs border on

converge and border on both are verbs.

Word	Noun	Adjective	Verb	Adverb
converge	No	No	Yes	No
border on	No	No	Yes	No

As verbs, border on is a hypernym of converge; that is, border on is a word with a broader meaning than converge:

converge: approach a limit as the number of terms increases without limit
border on: come near or verge on, resemble, come nearer in quality, or character

Other hypernyms of converge include approach.

converge (verb)	border on (verb)
come together so as to form a single product	come near or verge on, resemble, come nearer in quality, or character
move or draw together at a certain location
be adjacent or come together
approach a limit as the number of terms increases without limit