Answer by Alejandro Jenkins:
wrote an influential mathematical textbook, the Treatise on Fluxions (1742), which developed the calculus along Newtonian lines. There he made use of Taylor expansions about zero for various functions, giving due credit to . This had enough impact that people referred to that particular case as "Maclaurin expansions" (though I've only ever seen that term in passing comments in textbooks).
To complicate matters further, a distinguished modern mathematician has written that
"Integration had already been encountered with Archimedes, and differentiation with Pascal and Fermat; the connection between these operations was known to Barrow. What did Newton do in analysis? What was his main mathematical discovery? Newton invented Taylor series, the main instrument in analysis."
– V. I. Arnold, Huygens and Barrow, Newton and Hooke (Birkhäuser, 1990), p. 42
To complicate matters even further, many sources claim that the original discoverer of Taylor series was, who is older than Newton!
This often happens when you try to figure out who really "invented" a concept, because important ideas usually don't appear fully formed in the mind of a single person. And people often discover things that are later forgotten and re-discovered by someone else who has better luck, or who can see and advertise more clearly the importance of the idea.
Prof., at Caltech, likes to say that "we give credit to the person who discovers something so well that it's not necessary to discover it again."