Taylor series
A function written as an infinite sum of powers of x, with coefficients f^(n)(0) / n!. Partial sums give polynomial approximations.
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Analogy
A recipe that rebuilds a curve from its slope, bend, and higher-order behaviour at a single point.
Example
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.