1. The problem is to find the Taylor series formula for a function $f(x)$ around a point $a$. 2. The Taylor series of a function $f(x)$ at $x=a$ is given by the formula: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n$$ where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $a$, and $n!$ is the factorial of $n$. 3. Important rules: - The function must be infinitely differentiable at $a$. - The series approximates $f(x)$ near $x=a$. 4. To find the Taylor series: - Compute derivatives $f'(a), f''(a), f^{(3)}(a), \ldots$ - Substitute into the formula above. 5. Example: For $f(x) = e^x$ at $a=0$, all derivatives are $e^0=1$, so $$e^x = \sum_{n=0}^\infty \frac{1}{n!} x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ This formula allows you to approximate functions using polynomials around a point $a$.