Owl: Push a Taylor polynomial to infinite degree and you get a Taylor series. Centered at 0, it's called a Maclaurin series — not a different creature, just Taylor with a = 0.
What infinite series secretly is sin x? You'll recognize a pattern of odd powers with alternating signs. Once you know the series for eˣ, sin x, and cos x, you can manipulate them — substitute, differentiate, multiply — like algebraic clay.
Caveat: the series equals the function only where it converges to that function — not automatically on all of ℝ for every f. We'll write core series, recognize them, and manipulate carefully.