Taylor Series

Review Concepts

• Sequences
• Convergence
• Infinite series
• The sequence of partial sums of an infinite series
• Power series

New Concept: Taylor Series

• Think of Taylor series as a special kind of power series, where the sequence of partial sums are meant as better and better approximations of some other function.
• The Taylor Series is derived from the function.

Definition from

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series

$f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots.$

which can be written in the more compact sigma notation as

$\sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}$

where n! denotes the factorial of n and ƒ ⁽ⁿ⁾(a) denotes the nth derivative of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and Template:Nowrap and 0! are both defined to be 1. In the case that Template:Nowrap, the series is also called a Maclaurin series.