Taylor expansions

Japanese school year: University year 1

What you learn

You will learn how to approximate functions locally with polynomials using higher-order derivatives, as well as power series representations through Taylor and Maclaurin expansions. This approach is widely applied in numerical computation and physical modeling. A solid grasp of higher-order differentiation and series convergence is essential.

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Key points

This Taylor series formula expresses a smooth function $f(x)$ as a polynomial around a reference point $x = a$ . Here $f^{(n)}(a)$ is the $n$ -th derivative at $a$ , and $n!$ is $n$ factorial.

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n

This Maclaurin series formula expands a function $f(x)$ around $x = 0$ . Use it to approximate complicated functions near zero using powers of $x$ .

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n

This formula expresses the exponential function $e^x$ as an infinite sum of powers of $x$ . It holds true for any real number $x$ , where $n!$ is the factorial of $n$ .

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

This formula expresses $\sin x$ as a series containing only odd powers ( $x, x^3, x^5, \dots$ ) with alternating signs. It is valid for all real numbers $x$ .

\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}

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Practice 1Coming soon