Taylor and Maclaurin Series

Polynomial Approximations of Functions

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Taylor and Maclaurin series let us represent complicated functions as infinite sums of polynomial terms. This matters because polynomials are easier to evaluate, differentiate, and integrate than many other functions. Near a chosen point, a polynomial can closely match the behavior of the original function. These series are a major tool in calculus, physics, engineering, and numerical computation.

A Taylor series is built from the derivatives of a function at a specific center $a$ , while a Maclaurin series is the special case centered at $0$ . Each added term usually improves the approximation near the center, especially for smooth functions like $e^x$ , $\sin x$ , and $\cos x$ . The difference between the true function and the partial polynomial is called the remainder or error. Understanding where the approximation works well is just as important as writing the series itself.

Key Facts

Taylor series about $x = a$ : $f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$
Maclaurin series is the Taylor series with $a = 0$ : $f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots$
A degree $N$ Taylor polynomial is $T_N(x) = \sum_{n = 0}^{N} \frac{f^{(n)}(a)}{n!} (x - a)^n$

Vocabulary

Taylor series: An infinite series that represents a function using its derivatives at a chosen center point.
Maclaurin series: A Taylor series centered specifically at x = 0.
Taylor polynomial: The finite partial sum of a Taylor series used to approximate a function.
Center: The value a around which the Taylor series is constructed and where the approximation is most accurate.
Remainder: The difference between the actual function value and the value predicted by a Taylor polynomial.

Common Mistakes to Avoid

Using a Maclaurin series when the problem is centered at $x = a$ , because the powers must be written in terms of $(x - a)$ for a general Taylor series.
Forgetting the factorial in the denominator, which gives coefficients that are too large and makes the series incorrect.
Assuming the approximation is equally accurate for all x, even though Taylor polynomials are usually most accurate near the center and can become poor farther away.
Dropping alternating signs in series like sin x or cos x, which changes the function completely and leads to wrong values.

Practice Questions

1 Find the first four nonzero terms of the Maclaurin series for $e^x$ , then use them to approximate $e^{0.5}$ .
2 Write the degree 5 Maclaurin polynomial for sin x and use it to estimate sin(0.2).
3 Explain why a Taylor polynomial centered at x = 0 usually gives a better approximation near x = 0 than far from x = 0.