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Taylor & Maclaurin Series cheat sheet - grade 11-12

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Calculus Grade 11-12

Taylor & Maclaurin Series Cheat Sheet

A printable reference covering Taylor polynomials, Maclaurin series, common series, convergence, and error bounds for grades 11-12.

TopicsCalculusFunctionsDerivativesIntegralsGraphsStudy Guide
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Taylor and Maclaurin series let students represent complicated functions using polynomials near a chosen center. This cheat sheet helps organize the formulas, notation, and common series that appear in calculus problems. It is useful for approximating values, analyzing functions, and preparing for exams involving power series.

The main idea is that a function can be built from its derivatives at a point using powers of x−ax-a. A Maclaurin series is the special case of a Taylor series centered at a=0a=0. Students should know the general formulas, several common expansions, and how to check where a series converges.

Key Facts

  • The Taylor series for f(x)f(x) centered at x=ax=a is ∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.
  • The Maclaurin series is the Taylor series centered at a=0a=0, so f(x)=∑n=0∞f(n)(0)n!xnf(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n when the series converges to f(x)f(x).
  • The nnth Taylor polynomial is Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x)=\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k.
  • The geometric series formula is 11−x=∑n=0∞xn\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n for ∣x∣<1|x|<1.
  • The exponential series is ex=∑n=0∞xnn!e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!} for all real xx.
  • The sine and cosine series are sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} and cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}.
  • The Lagrange error bound is ∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)|\le \frac{M}{(n+1)!}|x-a|^{n+1} when ∣f(n+1)(t)∣≤M|f^{(n+1)}(t)|\le M between aa and xx.
  • The ratio test checks convergence using L=lim⁡n→∞∣an+1an∣L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|, where the series converges if L<1L<1 and diverges if L>1L>1.

Vocabulary

Taylor Series
An infinite power series that represents a function using its derivatives at a center aa.
Maclaurin Series
A Taylor series centered at a=0a=0.
Taylor Polynomial
A finite polynomial approximation Pn(x)P_n(x) made from the first n+1n+1 terms of a Taylor series.
Radius of Convergence
The distance RR from the center where a power series converges for ∣x−a∣<R|x-a|<R.
Interval of Convergence
The set of all xx-values where a power series converges, including any endpoints that work.
Remainder
The error term Rn(x)=f(x)−Pn(x)R_n(x)=f(x)-P_n(x) between the actual function value and the Taylor polynomial approximation.

Common Mistakes to Avoid

  • Forgetting the factorial in f(n)(a)n!(x−a)n\frac{f^{(n)}(a)}{n!}(x-a)^n is wrong because each coefficient must be divided by n!n!.
  • Using xnx^n instead of (x−a)n(x-a)^n for a Taylor series centered at a≠0a\ne 0 is wrong because the powers must measure distance from the center.
  • Assuming the radius of convergence includes the endpoints is wrong because endpoints must be tested separately.
  • Mixing up the sine and cosine series is wrong because sin⁡x\sin x uses odd powers and cos⁡x\cos x uses even powers.
  • Stopping after finding a Taylor polynomial and calling it the exact function is wrong because a polynomial is usually an approximation unless the remainder goes to 00.

Practice Questions

  1. 1 Find the degree 33 Maclaurin polynomial for exe^x.
  2. 2 Use the Maclaurin series for sin⁡x\sin x to approximate sin⁡(0.2)\sin(0.2) using terms through x3x^3.
  3. 3 Find the radius of convergence of ∑n=0∞(x−2)n3n\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n}.
  4. 4 Explain why endpoint testing is needed after the ratio test gives a radius of convergence.

Understanding Taylor & Maclaurin Series

A Taylor polynomial is a local model. It is designed to copy a function's value, slope, curvature, and higher changes at one chosen location. The constant term matches the height of the graph there.

The linear term matches its slope. The quadratic term accounts for bending. Each extra term forces one more derivative of the polynomial to agree with the function at the center.

This explains why the approximation is usually strongest near that point. Farther away, powers of the distance from the center grow, so small differences in later terms can become important. The factorial in each denominator keeps higher degree terms from becoming too large too quickly.

A finite Taylor polynomial and an infinite Taylor series are not the same object. A polynomial gives an estimate using a fixed number of terms. A series is the result hoped for when terms continue forever.

For many familiar functions, adding terms improves the result within a useful interval. Still, convergence alone needs care. A series can converge to a number that is not the original function at some points.

Some functions have derivatives of every order yet are not fully described by their Taylor series away from the center. In school problems, use the stated interval of convergence rather than assuming an expansion works everywhere.

The radius of convergence measures how far from the center a power series can be trusted to converge. The ratio test often finds this radius because it compares the size of neighboring terms when the degree becomes large. After finding the interval, test each endpoint separately.

The ratio test commonly gives no answer at an endpoint. A geometric type series is especially important because many other expansions come from it through substitution, differentiation, or integration.

Replacing its input with a simple expression can create a new series. Students should carefully track the new restriction on the input before using that result.

Error bounds turn an approximation into a justified estimate. First choose a center close to the value needed. Then choose enough terms so the error bound is smaller than the required tolerance.

To apply the bound, find a number that is at least as large as the next derivative on the entire interval between the center and the target value. This maximum matters because checking only one endpoint can give an unsafe bound. Series approximations appear in calculators, computer graphics, engineering models, and physics simulations when direct calculations are slow or difficult.

In class, pay attention to the center, the degree, the convergence interval, and the error. Those four details determine whether an answer is useful.