Maclaurin Series

College/University

Definition

The name for a power series in x for a function in the form of f(x). It helps provide an approximation near the origin for this function and allows you to calculate a sum that's usually uncomputable. A Maclaurin Series is a special case of a Taylor series.

Worked examples

\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\)

The Maclaurin series for the exponential function, centered at the origin.

\(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\)

Only odd powers appear, with alternating signs, matching sine's odd symmetry.

\(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\)

Only even powers appear with alternating signs, reflecting cosine's even symmetry.

Common mistakes

\(\sin x = x - \frac{x^2}{2!} + \frac{x^3}{3!} - \cdots\) → \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\) Sine uses only odd powers; even powers belong to cosine.
The Maclaurin series for \(f(x)\) is centered at \(x = a\) → A Maclaurin series is always centered at \(x = 0\) A Taylor series centered at any other point \(a \ne 0\) is not a Maclaurin series.
\(f(x) = \sum_{n=0}^{\infty} f(n) \frac{x^n}{n!}\) → \(f(x) = \sum_{n=0}^{\infty} f^{(n)}(0) \frac{x^n}{n!}\) Use the nth derivative evaluated at zero, not the function value at n.

Where you'll use it next

You'll use Maclaurin series to approximate transcendental functions in calculus and differential equations, evaluate limits with L'Hôpital's rule alternatives, solve power-series differential equations, and perform numerical computations in engineering and physics.

Found in 1 StudyPug lesson

Taylor and maclaurin series

Calculus 2

In this lesson, we will learn that most functions can be expressed as a Taylor Series. These are power series with a special form, and are centred at a point. If it is centred at 0, then it is called a Maclaurin Series. All of these series require the n'th derivative of the function at point a. We will first apply the Taylor Series formula to some functions. You may notice that trying to find a Taylor Series of a polynomial will just give us back the same polynomial, and not a power series. Then we will learn how to manipulate some of the formulas for some harder functions. Lastly, we will find the Taylor series for sine and cosine, which will requires us to recognize some patterns.

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