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Calculus II (Integration & Series) Unit 9: Power series & Taylor polynomials

Power series turn functions into infinite polynomials. Taylor polynomials approximate functions near a point, with a bound on the error.

Covers: Taylor/Maclaurin series, radius & interval of convergence, Lagrange error bound

Study guide

A power series centered at a is Σ cₙ(x − a)ⁿ. It converges on an interval around a with a radius of convergence R: it converges absolutely for |x − a| R. Find R with the ratio test, then test each endpoint separately to find the interval of convergence. Inside that interval, a power series can be differentiated and integrated term by term.

A Taylor series represents a function as a power series with coefficients cₙ = f⁽ⁿ⁾(a)/n!. A Maclaurin series is a Taylor series centered at 0. Key series to memorize: eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; 1/(1 − x) = 1 + x + x² + … for |x| < 1. Build new series by substituting (replace x with −x² in eˣ), multiplying by powers of x, differentiating or integrating known ones.

A Taylor polynomial Pₙ(x) keeps the first terms and approximates the function near the center. The Lagrange error bound says the error is at most M·|x − a|ⁿ⁺¹/(n + 1)!, where M bounds the (n + 1)th derivative between a and x. For alternating series, the error is at most the first omitted term. These bounds tell you how many terms you need for a given accuracy.

Differentiating or integrating a power series term by term keeps the same radius of convergence, but convergence at the endpoints can change, so recheck them. Also know ln(1 + x) = x − x²/2 + x³/3 − … for −1 < x ≤ 1 and arctan x = x − x³/3 + x⁵/5 − … for −1 ≤ x ≤ 1.

When f is defined by a differential equation, find f′(a), f″(a) and so on by repeatedly differentiating the equation (with the chain rule on y-terms) and substituting the known values. To write the general term, spot the patterns in signs, (−1)ⁿ, in powers and in factorials. For example, the nth term of the series for sin x is (−1)ⁿx^(2n+1)/(2n + 1)!.

Key ideas

• A power series Σ cₙ(x − a)ⁿ converges on an interval centered at a; check endpoints separately.
• Find the radius of convergence with the ratio test.
• Taylor coefficient: cₙ = f⁽ⁿ⁾(a)/n!.
• Build new series by substituting, differentiating or integrating known ones.
• Lagrange error bound limits how far the Taylor polynomial is from the function.

Formulas and rules

Common mistakes

• Forgetting to test the endpoints of the interval of convergence.
• Mixing up the n! in the Taylor coefficient.

Words to know

Taylor coefficient formula

cₙ = f⁽ⁿ⁾(a)/n!

Maclaurin series for eˣ

1 + x + x²/2! + x³/3! + …

Maclaurin series for sin x

x − x³/3! + x⁵/5! − …

Series for 1/(1 − x)

1 + x + x² + …, for |x| < 1

Radius of convergence

R in |x − a| < R; find it with the ratio test.

Lagrange error bound

|Rₙ(x)| ≤ M·|x − a|ⁿ⁺¹/(n + 1)!

Power series

Σ cₙ(x − a)ⁿ centered at a.

Interval of convergence

The x-values where a power series converges; test the endpoints separately.

Taylor series

Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ

Maclaurin series

A Taylor series centered at 0.

Maclaurin series for cos x

1 − x²/2! + x⁴/4! − …

Taylor polynomial

Pₙ(x): the first terms of a Taylor series, approximating f near a.

Alternating series error bound

The error is at most the first omitted term.

Sample practice questions

31 more questions like these, from level 1 to 5, are in the practice set.