Question 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.
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Write Maclaurin series for eˣ, sin x, and cos x Recognize known series in disguise Manipulate series by substitution and termwise operations Relate Maclaurin to Taylor and respect domains of equality
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What series is secretly sin x? Maclaurin = Taylor at a = 0
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