Taylor Series: Approximating Functions with Polynomials

The idea: match the derivatives

Polynomials are the easiest functions to work with. You can evaluate them using only addition and multiplication, and differentiating or integrating them is routine. Functions such as $e^x$, $\sin x$ and $\ln x$ are much harder to compute by hand. The idea of a Taylor series is to find a polynomial that behaves like the function near a chosen point $a$.

The best line through a point is the tangent line, which matches the function's value and its first derivative there. A parabola can also match the second derivative, so it follows the curve's bending. A cubic can match the third derivative as well, and so on. Each extra matched derivative makes the polynomial hug the function over a wider stretch around the point. Continuing forever gives the Taylor series.

The Taylor series formula

Requiring the polynomial's $n$th derivative at $a$ to equal $f^{(n)}(a)$ forces the coefficient of $(x - a)^n$ to be $\frac{f^{(n)}(a)}{n!}$. The factorial appears because differentiating $(x - a)^n$ $n$ times produces $n!$. The full series is:

When $a = 0$, the series is called a Maclaurin series, and the powers are simply $x^n$. The polynomial formed by the terms up to $(x - a)^n$ is the Taylor polynomial of degree $n$, written $P_n(x)$. The first two terms alone are the tangent line approximation, also called linearisation.

Building the series for e to the x

Setting $x = 1$ gives a famous formula for the constant itself: $e = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$, which reaches 2.71828 after a dozen terms because the factorials grow so quickly.

Series worth knowing

These Maclaurin series appear constantly, and most others can be built from them by substitution, multiplication, differentiation or integration:

Sine contains only odd powers, matching the fact that it is an odd function, and cosine only even powers, matching its symmetry. Differentiating the sine series term by term gives the cosine series, exactly as the derivative of $\sin x$ should be $\cos x$.

As a check, $\sin 0.1 \approx 0.1 - \frac{0.001}{6} = 0.0998333$, while the true value is $0.0998334$. Two terms already give six correct decimal places, which is why calculators can evaluate trigonometric functions so quickly with short polynomials.

Expanding around a point other than zero

Choosing $a$ near the input you care about makes the approximation converge much faster. To estimate $\sqrt{4.1}$, expand $\sqrt{x}$ around $a = 4$, where the square root is easy.

How large is the error?

Taylor's theorem gives the error of a Taylor polynomial. If $|f^{(n+1)}| \le M$ between $a$ and $x$, then the remainder after the degree $n$ term satisfies:

For $e^{0.5}$ with $n = 3$, the fourth derivative is $e^t \le e^{0.5} < 2$ on the interval, so the error is at most $\frac{2(0.5)^4}{24} \approx 0.0052$. The actual error, $0.0029$, is within that bound. The bound is useful in reverse as well: it tells you how many terms you need to guarantee a given accuracy before you start computing.

For alternating series such as those for sine and cosine, there is an even simpler estimate: the error is no larger than the first omitted term. The ideas behind that rule are covered in series convergence tests.

Where the series converges

A Taylor series does not always equal its function everywhere. The geometric series for $\frac{1}{1 - x}$ works only for $|x| < 1$; at $x = 2$ the left side is $-1$ while the terms of the series grow without bound. The set of $x$ values where a series converges is an interval centred at $a$, and half its width is the radius of convergence. It is usually found with the ratio test.

The series for $e^x$, $\sin x$ and $\cos x$ converge for every real $x$. The series for $\ln(1 + x)$ converges for $-1 < x \le 1$; at $x = 1$ it gives the striking result $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$, although it converges very slowly.

What Taylor series are used for

• Evaluating functions: calculators and software use polynomial approximations closely related to Taylor polynomials.
• Limits: replacing $\sin x$ by $x - \frac{x^3}{6} + \cdots$ shows that $\lim_{x \to 0}\frac{\sin x - x}{x^3} = -\frac{1}{6}$ without L'Hopital's rule.
• Integrals with no elementary antiderivative: integrating the series for $e^{-x^2}$ term by term gives $\int_0^1 e^{-x^2}\,dx \approx 1 - \frac{1}{3} + \frac{1}{10} - \frac{1}{42} + \frac{1}{216} \approx 0.7475$, close to the true $0.7468$.
• Physics: the small angle approximation $\sin\theta \approx \theta$ used for pendulums is the first term of the sine series.
• Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, which follows by comparing the three series.

The Taylor series calculator expands any function to a chosen order around any point and shows each derivative. Derivative skills are essential here, so derivative rules explained is a useful companion.

A recipe for any Taylor polynomial

1. Choose the centre $a$: a point near where you need values, where the function and its derivatives are easy to evaluate.
2. Differentiate the function as many times as the degree you want, and simplify each derivative.
3. Evaluate each derivative at $a$.
4. Divide the $n$th value by $n!$ to get each coefficient.
5. Write the polynomial with powers of $(x - a)$, not powers of $x$, unless $a = 0$.
6. Check the first coefficient: $P_n(a)$ must equal $f(a)$, and the slope of $P_n$ at $a$ must equal $f'(a)$.

It is often faster to build a new series from a known one than to differentiate repeatedly. For instance, replacing $x$ by $-x^2$ in the series for $e^x$ gives $e^{-x^2} = 1 - x^2 + \frac{x^4}{2} - \cdots$ immediately, and multiplying the sine series by $x$ gives the series for $x\sin x$.

Seeing the approximations on a graph

Plot $\cos x$ together with its Taylor polynomials of degree 2, 4 and 6 and a clear pattern appears. The degree 2 polynomial, a downward parabola, matches the cosine curve well near zero but drifts away beyond about one unit. The degree 4 polynomial stays close for longer, and the degree 6 polynomial follows the curve through most of its first hump. Every polynomial eventually shoots off to infinity, because no polynomial can oscillate forever, but adding terms pushes that departure further and further out, which is exactly what convergence of the series means in practice.

This picture is the best way to build intuition: the approximation is excellent near the centre and gets worse with distance. The graphing calculator can plot a function and several of its Taylor polynomials together so you can watch them converge.

Frequently asked questions