## Taylor Series A Taylor series is a way to represent a function as an infinite series of terms, each involving higher powers of the distance from a chosen center point $c$. It reconstructs the function using its value and all its derivatives at $c$. The general formula is: $ g(x) = g(c) + g^\prime(c)(x-c) + \frac{g^{\prime\prime}(c)}{2!}(x-c)^2 + \frac{g^{\prime\prime\prime}(c)}{3!}(x-c)^3 + \dots $ This series approximates $g(x)$ at points near $c$, and as more terms are added, the approximation improves. Each term uses derivatives of increasing order, scaled by $(x-c)^n$ and normalized by $\frac{1}{n!}$ to reflect their influence. In summation form: $ g(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n$ ## Maclaurin Series A Maclaurin series is a special case of the Taylor series where the center of the series is at $c=0$. Its formula becomes: $ g(x) = g(0) + g^\prime(0)x + \frac{g^{\prime\prime}(0)}{2!}x^2 + \frac{g^{\prime\prime\prime}(0)}{3!}x^3 + \dots $ In summation form: $ g(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}\,x^n $ This series is often used when functions are symmetric or centered around the origin. ## Taylor Series Expansion When we know everything about a function $g$ at a single point, such as $c$, we can use the Taylor series to reconstruct the function around that point. By "everything" we mean, that we know the following properties: ^abca8c - Function value: $g(c)$ - All derivatives at $c$: $g^\prime(c), \:g^{\prime\prime}(c), \: \dots$ > [!note:] > This method is only valid for well-behaved functions. This means that they need to be continuous and differentiable as many times as we want.