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What is the coefficient of xⁿ in a Maclaurin series?
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All Flashcards in Topic 5.19
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5.19.18 cards
What is the coefficient of xⁿ in a Maclaurin series?
f⁽ⁿ⁾(0) ÷ n! — the nth derivative evaluated at x = 0, divided by n!.
Write the general Maclaurin series of f(x).
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …
Why does the Maclaurin formula divide by n!?
Differentiating xⁿ exactly n times gives n!; dividing by n! makes the nth term contribute exactly f⁽ⁿ⁾(0) to the nth derivative at 0.
Maclaurin series of eˣ?
eˣ = 1 + x + x²/2! + x³/3! + … (every power, denominator n!).
Maclaurin series of sin x?
sin x = x − x³/3! + x⁵/5! − … (odd powers only, alternating signs).
Maclaurin series of cos x?
cos x = 1 − x²/2! + x⁴/4! − … (even powers only, alternating signs).
Maclaurin series of ln(1 + x)?
ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … (plain denominators, alternating signs).
Why does sin x contain only odd powers?
Every even derivative of sin x equals ±sin 0 = 0, so all even-power coefficients vanish.
5.19.28 cards
How do you find a Maclaurin series by substitution?
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! + ….
Maclaurin series of e^{x²}?
1 + x² + x⁴/2! + … = 1 + x² + x⁴/2 + … (substitute x² into eˣ).
Maclaurin series of sin(3x) (first two terms)?
3x − (3x)³/3! + … = 3x − (9/2)x³ + ….
How do you get the series of x·sin x?
Multiply the sin x series by x: x(x − x³/6 + …) = x² − x⁴/6 + ….
How do you use a Maclaurin series to find a 0/0 limit?
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).
Evaluate lim(x→0) (sin x − x)/x³.
sin x − x = −x³/6 + …, so dividing by x³ gives −1/6 as x → 0.
Evaluate lim(x→0) (1 − cos x)/x².
1 − cos x = x²/2 − …, so the limit is 1/2.
When multiplying two series, which terms do you keep?
Only terms up to the highest power the question requires; discard anything higher to save work.
Topic 5.19 study notes
Full notes & explanations for Maclaurin series (HL only)
Math AA exam skills
Paper structures, command terms & tips
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