Using known Maclaurin series
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Flip to reveal answersHow do you find a Maclaurin series by substitution?
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Question
How do you find a Maclaurin series by substitution?
Answer
Take a known series (eˣ, sin x, …) and replace every x by the new expression, e.g. x² into eˣ gives e^{x²} = 1 + x² + x⁴/2! + ….
Question
Maclaurin series of e^{x²}?
Answer
1 + x² + x⁴/2! + … = 1 + x² + x⁴/2 + … (substitute x² into eˣ).
Question
Maclaurin series of sin(3x) (first two terms)?
Answer
3x − (3x)³/3! + … = 3x − (9/2)x³ + ….
Question
How do you get the series of x·sin x?
Answer
Multiply the sin x series by x: x(x − x³/6 + …) = x² − x⁴/6 + ….
Question
How do you use a Maclaurin series to find a 0/0 limit?
Answer
Replace the function with its series; the leading terms cancel, divide by the matching power, then let x → 0 (the constant term is the limit).
Question
Evaluate lim(x→0) (sin x − x)/x³.
Answer
sin x − x = −x³/6 + …, so dividing by x³ gives −1/6 as x → 0.
Question
Evaluate lim(x→0) (1 − cos x)/x².
Answer
1 − cos x = x²/2 − …, so the limit is 1/2.
Question
When multiplying two series, which terms do you keep?
Answer
Only terms up to the highest power the question requires; discard anything higher to save work.
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Topic 5.19 hub
Maclaurin series (HL only)
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