Short on time? This page is the recap. For the reasons, the graph and the worked examples, use the full study guide. BC-only material. Prerequisites (geometric series 10.2, radius and interval of convergence 10.13, Maclaurin series 10.14) are on the Calculus BC roadmap.
Recap
- Start from 1/(1 − x), eˣ, sin x or cos x, and change the series the way the function changes.
- A power series with a positive radius is the Taylor series of its sum, so any correct route gives the same series. The coefficient of xⁿ is f⁽ⁿ⁾(0)/n!.
- Inside the interval of convergence you can differentiate and integrate term by term.
Key relationships
| Operation | Example | Interval |
|---|---|---|
| Geometric form a/(1 − r) | 2/(5 − x) = (2/5) · 1/(1 − x/5) = Σ 2xⁿ/5ⁿ⁺¹ | |x/5| < 1: −5 < x < 5 (geometric endpoints always diverge) |
| Substitute | e^(−x²) = Σ (−1)ⁿ x^(2n)/n! | All real x |
| Multiply or divide by xᵏ | (1 − cos x)/x² = 1/2 − x²/24 + x⁴/720 − … | Same as the original series |
| Differentiate | 1/(1 − x)² = Σ from n = 1 of n xⁿ⁻¹ | Same radius; retest endpoints |
| Integrate | ln(4 + x) = ln 4 + Σ (−1)ⁿ xⁿ⁺¹/((n + 1)4ⁿ⁺¹) | −4 < x ≤ 4 (gained x = 4) |
Assumptions
- Series are about x = 0 unless a centre c is given; then use powers of (x − c).
- “Valid” means the series converges and equals the function there.
Mistakes to avoid
- No 1 in front. Rewrite 5/(3 + x) as (5/3) · 1/(1 + x/3) first.
- Substituting outside the powers. sin(3x) has (3x)³ = 27x³.
- Old interval after a substitution. Put the new r into |r| < 1.
- Missing constant after integrating. Check the value at the centre.
- Assuming endpoints carry over. Only the radius is guaranteed.
- Coefficient used as the derivative. f⁽ⁿ⁾(0) = n! × (coefficient of xⁿ).
- Using a series outside its interval. The function may exist there; the series does not equal it.
Quick self-check
- Write the series for 1/(1 − x³) and its interval. (1 + x³ + x⁶ + x⁹ + … = Σ x^(3n), for −1 < x < 1)
- Write the first three nonzero terms of the series for (sin x)/x. (1 − x²/6 + x⁴/120)
- Find the coefficient of x⁵ in the series for x e^(2x). (2⁴/4! = 2/3)
- Use x/(1 − x)² = Σ n xⁿ to find Σ from n = 1 of n/3ⁿ. (x = 1/3 gives 3/4)
Next: practice questions.