Taylor Series and Approximation
A power series is a polynomial with infinitely many terms, and Taylor's formula tells you exactly which coefficients make it copy a function at a point.
Power series basics
A power series centered at a has the form sum of c_n times (x - a) to the n, and it converges inside a radius R and diverges outside it. For example, the geometric-style series sum x^n has R equal to 1, working only between -1 and 1. Endpoints often need a separate check, since the interior test says nothing there.
\sum_{n=0}^{\infty} c_n (x-a)^nTaylor coefficients
Taylor's insight is that the coefficient matching a function f at the point a is the nth derivative of f at a, divided by n factorial. The resulting polynomial agrees with f in value and in every derivative at that point, which is the strongest possible local match. When a is 0 the series is called a Maclaurin series.
c_n = \frac{f^{(n)}(a)}{n!}Famous expansions
Three expansions cover most homework: e to the x is sum x to the n over n factorial, sine is x minus x cubed over 6, and 1 over (1 - x) is sum x to the n for |x| below 1. Try approximating e with the first three terms of the exponential series: 1 + 1 + 1/2 gives 2.5, already close to the true value near 2.718.
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}Flashcard
Taylor coefficient from the nth derivative
Check your understanding
What is the radius of convergence of the power series whose nth term is x to the n divided by n factorial?