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Taylor series

Approximating any smooth function by polynomials built from its derivatives at one point — the most used approximation in science and engineering.

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Topics

Taylor polynomial

The polynomial of degree $n$ that matches $f$ and its first $n$ derivatives at a point $a$ . Degree 1 is the tangent line, degree 2 adds curvature; the higher the degree, the wider the region where it is a good approximation. At $a = 0$ it is called a Maclaurin polynomial.

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Taylor's theorem and the remainder

The exact size of the approximation error: $f(x) - T_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - a)^{n+1}$ for some $\xi$ between $a$ and $x$ . It is how numerical analysts certify accuracy.

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Power series

"Polynomials of infinite degree" $\sum_k c_k (x - a)^k$ . Inside their interval of convergence they can be added, multiplied, differentiated and integrated term by term — like polynomials.

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Radius of convergence

A power series converges for $|x - a| < R$ and diverges for $|x - a| > R$ . Surprisingly, $R$ is the distance from $a$ to the nearest singularity in the complex plane: the series of $\frac{1}{1 + x^2}$ at 0 has $R = 1$ because of the poles at $\pm i$ , although the function is perfectly smooth on $\R$ .

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Taylor and Maclaurin series

Let the order go to infinity: $f(x) = \sum_{k\ge0}\frac{f^{(k)}(a)}{k!}(x - a)^k$ whenever the remainder tends to 0. Functions equal to their Taylor series are called analytic; $e^x$ , $\sin$ , $\cos$ are, on all of $\R$ .

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Where this area leads in computing

Topics

Taylor polynomial

Taylor's theorem and the remainder

Power series

Radius of convergence

Taylor and Maclaurin series

Where this area leads in computing

λ Scientific computing and algorithms ★★★★★

ℒ AI and machine learning ★★★★★

⚛ Physics and simulation ★★★★★

ψ Quantum computing and physics ★★★★★