A function written as a polynomial: Imagine zooming in on a curve near x = 0. A Maclaurin series copies the function there by matching its height, slope, bend, and every higher derivative at x = 0 — building an endless polynomial that agrees with the curve.
The coefficient of xⁿ is f⁽ⁿ⁾(0) ÷ n!: the nth derivative measured at 0, then shared out by n! so each new term doesn't disturb the lower ones.
Why divide by n!?: Differentiate xⁿ exactly n times and you get n! (a constant). Dividing the coefficient by n! cancels that, so when you set x = 0 the nth term contributes exactly f⁽ⁿ⁾(0) to the nth derivative — and nothing to the lower ones. That is what makes every derivative match at x = 0.
IB-style question — find the series from scratch
Let f(x) = e2x.
Find the Maclaurin series of f up to and including the term in x³.
Step by step
- List the derivatives, then read each one at x = 0.
- Evaluate every derivative at 0 (e⁰ = 1).
- Drop them into the formula, dividing by the factorials.
- Simplify 4/2! = 2 and 8/3! = 4/3.
Final answer
e2x = 1 + 2x + 2x² + (4/3)x³ + … (the same as eˣ with x replaced by 2x).
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Four series worth knowing cold: These four come up again and again. eˣ uses every term; sin x keeps only odd powers; cos x keeps only even powers; ln(1 + x) has alternating signs and plain whole-number denominators (no factorials).
Learn them — then most exam questions are just substitution or multiplication, not differentiating from scratch.
IB-style question — derive sin x to confirm the pattern
Let f(x) = sin x.
Use the Maclaurin formula to find the first three non-zero terms, and explain why all the even powers vanish.
Step by step
- The derivatives of sin x cycle in fours.
- Read each at x = 0 (sin 0 = 0, cos 0 = 1).
- Every EVEN derivative gives 0, so all even powers drop out — that's why only odd powers survive.
- Use the surviving (odd) terms.
Final answer
sin x = x − x³/3! + x⁵/5! − … ; even powers vanish because every even derivative is ±sin 0 = 0.
IB-style question — derive ln(1 + x)
Find the Maclaurin series of f(x) = ln(1 + x) up to the term in x³.
Step by step
- Differentiate repeatedly.
- Read each at x = 0 (and f(0) = ln 1 = 0).
- Apply the formula, dividing by factorials.
- Simplify 1/2! = 1/2 and 2/3! = 1/3 — the factorials cancel to plain denominators.
Final answer
ln(1 + x) = x − x²/2 + x³/3 − … (plain denominators 1, 2, 3, …).