Math · Calculus II · Concept

Taylor polynomials and Taylor series

A Taylor polynomial of degree n matches a function’s value and first n derivatives at a center a. Letting n grow gives the Taylor series; centered at 0 it is a Maclaurin series. Taylor’s remainder bounds the error of the polynomial, and the ratio test finds the interval where the series converges.

Match more than the value

A constant matches the height at a; adding a linear term also matches the slope, giving the tangent line; a quadratic term matches the curvature. Powers of x − a keep these requirements separate, because every higher power and its lower derivatives vanish at a.

P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} {(x - a)}^{k}

Where the factorial comes from

Differentiating (x − a)ᵏ exactly k times gives k!, so dividing by k! makes the kth derivative of the polynomial at a equal to f⁽ᵏ⁾(a).

Maclaurin series to know

Centered at 0: eˣ = 1 + x + x²/2! + x³/3! + ⋯ for every x; sin x = x − x³/3! + x⁵/5! − ⋯; cos x = 1 − x²/2! + x⁴/4! − ⋯; and 1/(1 − x) = 1 + x + x² + ⋯ only when |x| < 1.

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

Bound the error

If the (n + 1)st derivative stays within M in size between a and x, Taylor’s remainder gives the bound below. For an alternating series, such as sin x at small x, the first omitted term bounds the error as well. A sampled maximum error, as a graph shows, is useful feedback but not a proven bound.

∣ f (x) - P_{n} (x) ∣ \leq \frac{M ∣ x - a ∣^{n + 1}}{(n + 1)!}

Where a series converges

A power series in x − a converges on an interval centered at a. The ratio test finds its radius; the endpoints need separate tests, because there the ratio test decides nothing.

Common mistakes

Forgetting the factorials: the coefficient of (x − a)ᵏ is f⁽ᵏ⁾(a)/k!, not f⁽ᵏ⁾(a).
Assuming the center must be zero: a Taylor polynomial can be centered at any a where f has the derivatives.
Using a series outside its interval of convergence: 1 + x + x² + ⋯ does not equal 1/(1 − x) at x = 2.
Treating a sampled maximum error as a proven bound.

Key terms

Taylor polynomial: The polynomial of degree n that matches a function’s value and first n derivatives at a center a: Pₙ(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ/k! for k = 0 to n.
Taylor series: The power series Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n! built from all of a function’s derivatives at a. Where it converges to the function, it represents the function as an infinite polynomial.
Maclaurin series: A Taylor series centered at 0. It writes f(x) as f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …, built from the function’s derivatives at x = 0.
Power series: A series of powers of (x − a), Σcₙ(x − a)ⁿ, centered at a. Whether it converges depends on x: it converges inside its radius of convergence, and each endpoint must be checked separately.
Radius of convergence: The distance R from the center within which a power series converges, absolutely. R can be 0 or ∞; at exactly distance R, test each endpoint on its own.
Interval of convergence: All the x-values where a power series converges: the interval inside the radius of convergence, plus whichever endpoints pass their own tests.
Factorial: For a positive integer n, n! is the product of the integers from 1 through n; 0! is defined as 1. Factorials count arrangements of distinct objects, and the calculator limits factorial input to integers from 0 through 170.

Work through an example

Use the degree-two Taylor polynomial of eˣ at 0 to approximate e^0.2, and bound the error.

Approximate e^0.2 with a Taylor polynomial →

Find the Maclaurin series of sin x →

Find the interval of convergence →

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