# Taylor series A function written as an infinite sum of powers of x, with coefficients f^(n)(0) / n!. Partial sums give polynomial approximations. Maths (university) · Part 07: Sequences and series · Card 9 of 9 · Last updated 2026-10-19 ## Analogy A recipe that rebuilds a curve from its slope, bend, and higher-order behaviour at a single point. ## Example **e^x = 1 + x + x^2/2 + x^3/6 + ...** Take x = 0.1. 1 + 0.1 + 0.005 + 0.000167. = 1.105167. True e^0.1 = 1.105171. ## In practice Calculators use these to compute e^x, sin x and ln x. ## Practice ### Flashcards - **Write the first four terms of the Taylor series of e^x.** 1 + x + x^2/2 + x^3/6. ## Questions people ask ### Where does the Taylor series formula come from? Match a polynomial to the function at one point: its value, slope, and bend. This forces each coefficient to be the nth derivative at that point divided by n factorial. For e to the x every derivative at 0 is 1, giving 1 over n factorial. ### What is the difference between Taylor and Maclaurin series? A Maclaurin series is a Taylor series centred at 0. A Taylor series can be centred at any point a, using powers of x minus a. Maclaurin series are simpler, so they are used for e to the x, sin x and cos x. ## Related concepts - [Partial sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-2) - [Geometric series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-3) - [Divergence test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-4) - [p-series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-5) Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-9 Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-7-concept-9