# 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:

- 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). - 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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