1. Taylor Series

By M. Bourne

Our aim is to find a polynomial that gives us a good approximation to some function. (See why we want to do this in the Introduction.)

We find the desired polynomial approximation using the Taylor Series.

If we want a good approximation to the function in the region near `x = a`, we need to find the first, second, third (and so on) derivatives of the function and substitute the value of a. Then we need to multiply those values by corresponding powers of `(x − a)`, giving us the Taylor Series expansion of the function `f(x)` about `x = a`:

`f(x)` `~~f(a)+f’(a)(x-a)+(f’’(a))/(2!)(x-a)^2``+(f’’’(a))/(3!)(x-a)^3``+(f^("iv")(a))/(4!)(x-a)^4``+...`

Getting Lost?

See some background to why this sum converges to a polynomial in Infinite Geometric Series from an earlier chapter.

We can write this more conveniently using summation notation as:

`f(x)~~sum_(n=0)^oo(f^((n))(a))/(n!)(x-a)^n`

Conditions

In order to find such a series, some conditions have to be in place:

  1. The function `f(x)` has to be infinitely differentiable (that is, we can find each of the first derivative, second derivative, third derivative, and so on forever).
  2. The function `f(x)` has to be defined in a region near the value `x = a`.

Let's see what a Taylor Series is all about with an example.

Example - Expansion of ln x

Find the Taylor Expansion of `f(x) = ln x` near `x = 10`.

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