Short on time? This page is the recap. For the derivations, the partial-sum graph and worked examples, use the full study guide. BC-only material. Prerequisites (Taylor polynomials 10.11, radius and interval 10.13, geometric series 10.2) are on the Calculus BC roadmap.
Recap
- Taylor series about c: f(c) + f′(c)(x − c) + f″(c)(x − c)²/2! + … = Σ f⁽ⁿ⁾(c)(x − c)ⁿ/n!.
- Maclaurin series: the same with c = 0.
- Taylor polynomial Pₙ = the partial sum up to the (x − c)ⁿ term.
- To build a series from scratch: find several derivatives at c, spot the pattern, divide by n!, write the general term, then find the interval.
The four series to know
| f(x) | Series | Interval |
|---|---|---|
| eˣ | Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + … | all real x |
| sin x | Σ (−1)ⁿ x^(2n+1)/(2n + 1)! = x − x³/3! + x⁵/5! − … | all real x |
| cos x | Σ (−1)ⁿ x^(2n)/(2n)! = 1 − x²/2! + x⁴/4! − … | all real x |
| 1/(1 − x) | Σ xⁿ = 1 + x + x² + … (geometric, ratio x) | −1 < x < 1 |
Key relationships
| Idea | Fact | Example |
|---|---|---|
| Coefficient to derivative | f⁽ⁿ⁾(c) = n! · aₙ | a₅ = −7/120 gives f⁽⁵⁾(c) = −7 |
| Local behaviour | sign of a₁ gives increasing or decreasing; sign of a₂ gives concavity | a₁ = −3, a₂ = 2: decreasing, concave up |
| New centre | powers of (x − c) | eˣ = Σ e⁻¹(x + 1)ⁿ/n! about −1 |
| Substitution | replace x by the whole expression | e^(−2x) = Σ (−2x)ⁿ/n! |
| Recognising sums | match powers, factorials, signs | Σ 3ⁿ/n! = e³ |
Mistakes to avoid
- Leaving out the n! in the coefficient.
- Writing xⁿ instead of (x − c)ⁿ for a centre c ≠ 0.
- Swapping the sin and cos patterns (odd/even powers and factorials).
- Using 1/(1 − x) = Σ xⁿ when |x| ≥ 1.
- Treating a coefficient as the derivative itself.
- Forgetting to give the general term or the interval when asked.
Quick self-check
- First three nonzero terms of the Maclaurin series for cos x? (1 − x²/2 + x⁴/24)
- The Taylor series for f about x = 2 has (x − 2)⁴ coefficient 1/3. Find f⁽⁴⁾(2). (4! · 1/3 = 8)
- What is 1 − 1/3! + 1/5! − 1/7! + … exactly? (sin 1)
Next: practice questions.