Series Expansion: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Limits & Continuity
• Derivatives
• Applications of Derivatives
• Integration
• Applications of Integrals

Taylor Series

Σ f^(n)(a)/n! · (x-a)^n. Polynomial approximation around point a. Exact for polynomials.

Maclaurin Series

Taylor series around a=0. Σ f^(n)(0)/n! · x^n. Special case of Taylor.

Common Maclaurin Series

e^x = Σ x^n/n!, sin(x) = Σ (-1)^n x^(2n+1)/(2n+1)!, cos(x) = Σ (-1)^n x^(2n)/(2n)!

Radius of Convergence

Power series Σ aₙ(x-c)^n converges for |x-c|<R. Found using ratio test.

Interval of Convergence

Range of x values where series converges. May include endpoints (test separately).

Error in Taylor Polynomial

Remainder R_n(x) = f(x) - P_n(x). Bounded by Lagrange error formula.

Lagrange Error Bound

|R_n(x)| ≤ M|x-a|^(n+1)/(n+1)! where M = max|f^(n+1)| on interval.

Power Series Manipulation

Add, multiply, differentiate, integrate power series term-by-term within convergence radius.

Binomial Series

(1+x)^r = 1 + rx + r(r-1)x²/2! + ... Converges for |x|<1. Extends to non-integer r.

Unit 10 Summary

Taylor/Maclaurin series approximate functions. Convergence radius determined by ratio test. Error bounded by Lagrange formula.

Taylor Polynomial Definition

P_n(x) = Σ(k=0 to n) f^(k)(a)(x-a)^k/k!. Each coefficient forces the polynomial to match one more derivative of f at a.

Maclaurin Series

A Taylor series centered at a=0. The standard ones for e^x, sin x, cos x, and 1/(1-x) should be memorized rather than derived under time pressure.

Series for e^x

e^x = Σ x^n/n! for all real x, converging everywhere because factorial growth beats any fixed power.

Series for sin and cos

sin x = x - x³/3! + x⁵/5! - ... uses odd powers only; cos x = 1 - x²/2! + x⁴/4! - ... uses even powers. Both converge for all x.

Geometric Series as a Template

1/(1-x) = Σ x^n for |x|<1. Substituting -x², 2x, or similar expressions generates many other series instantly.

Series for ln(1+x)

ln(1+x) = x - x²/2 + x³/3 - ... for -1 < x ≤ 1, obtained by integrating the geometric series for 1/(1+x) term by term.

Series for arctan x

arctan x = x - x³/3 + x⁵/5 - ... for |x| ≤ 1, from integrating 1/(1+x²). At x=1 it gives the slowly converging Leibniz formula for π/4.

Binomial Series

(1+x)^m = 1 + mx + m(m-1)x²/2! + ... converges for |x|<1 and reduces to a finite expansion when m is a nonnegative integer.

Lagrange Error Bound

|R_n(x)| ≤ M|x-a|^(n+1)/(n+1)! where M bounds |f^(n+1)| on the interval between a and x.

Choosing M

M must bound the (n+1)st derivative on the whole interval, not just at the center; sine and cosine always allow M=1.

Alternating Bound vs Lagrange

If the Taylor series alternates with decreasing terms, the simpler alternating bound applies; otherwise use Lagrange.

Operations on Series

Series may be added, multiplied, substituted into, differentiated, and integrated term by term within the common interval of convergence.

Finding a Series by Substitution

To expand e^(-x²), substitute -x² into the e^u series rather than differentiating repeatedly — far faster and less error-prone.

Evaluating Limits with Series

Replacing functions by their first few Maclaurin terms resolves 0/0 limits without repeated L'Hôpital passes.

Approximating Integrals with Series

Functions like sin(x)/x or e^(-x²) with no elementary antiderivative can be integrated term by term to any desired accuracy.

Coefficient Extraction

The coefficient of (x-a)^n in a Taylor series equals f^(n)(a)/n!, so f^(n)(a) = n!·c_n — a common exam question in reverse.

Radius of Convergence of a Taylor Series

Determined by the ratio test on the coefficients; for real functions it often equals the distance to the nearest point where the function misbehaves.

Degree Needed for an Accuracy Goal

Set the error bound less than the target tolerance and solve for the smallest n, testing integer values since the inequality is rarely solvable in closed form.

Taylor Polynomials from a Table of Derivatives

AP problems frequently give f(a), f'(a), f''(a), f'''(a) and ask you to build P_3 directly, dividing each by the appropriate factorial.

Series and Function Identity

A Taylor series equals its function only where the remainder tends to zero; for e^x, sin x, and cos x this holds everywhere.