gen-define-a-taylor-series-and-give-one-short-worked-example-6a7b

Define a Taylor series and give one short worked example.

# Taylor Series ## Definition A **Taylor series** is an infinite sum that represents a function as a power series centered at a point *a*: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$ where *f*^(*n*)(*a*) is the *n*-th derivative of *f* evaluated at *x = a*. When *a = 0*, this is called a **Maclaurin series**. ## Worked Example **Find the Taylor series for** ***f(x) = e^x*** **centered at** ***a = 0***. **Solution:** First, find the derivatives: - *f(x) = e^x* → *f(0) = 1* - *f'(x) = e^x* → *f'(0) = 1* - *f''(x) = e^x* → *f''(0) = 1* - *f^(n)(x) = e^x* → *f^(n)*(0) = 1* for all *n* Substitute into the formula: $$e^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$ This series converges for all real *x*.