Taylor Series

Let $f$ be infinitely many times differentiable on $(a,b)$ and $c,x \in (a,b)$ .

If $\lim_{n\to \infty} R_n(x) = 0$ for $x \in (c-R, c+R) \subset (a,b)$ , then Taylor series of $f$ at $c$ is given by:

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

Procedure

Suppose a function $f$ is given and its Taylor series is required.

Differentiate $f$ repeatedly and find a general solution for $n$ -th derivative
Construct Taylor polynomial
Use root test or ratio test to find the range of convergence of the Taylor polynomial
Consider the endpoints of range of convergence to check if the Taylor polynomial converges
Construct the Taylor remainder
Find for which values of $x$ , the remainder converges to $0$

Range of convergence is $\mathbb{R}$ .

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Range of convergence is $(-1,1]$ .

\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots

Range of convergence is $\mathbb{R}$ .

\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

Range of convergence is $\mathbb{R}$ .

\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

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October 8, 2026 1 min read

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