1. The Taylor series is a way to represent a function as an infinite sum of terms calculated from the derivatives of the function at a single point. 2. Suppose we have a function $f(x)$ that is infinitely differentiable at a point $a$. The Taylor series of $f(x)$ around $a$ is given by: $$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$$ 3. Here, $f^{(n)}(a)$ denotes the $n$th derivative of $f$ evaluated at $a$, and $n!$ is the factorial of $n$. 4. The series approximates $f(x)$ near the point $a$. The more terms you include, the better the approximation. 5. For example, the Taylor series of $e^x$ at $a=0$ (also called the Maclaurin series) is: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^\infty \frac{x^n}{n!}$$ 6. Taylor series are useful in calculus, physics, and engineering to approximate complicated functions with polynomials, which are easier to work with. 7. Remember, the convergence of the Taylor series depends on the function and the point $a$; sometimes it converges only within a certain radius around $a$.