In this short video, I'm just going to give you

one more teaser about what the Taylor series is doing,

before we try and write down anything like a formal definition.

This will allow you to have a go at some graphical questions first,

which is much like how we approach learning differentiation at the start of this course.

Taylor series are also referred to as power series.

And this is because they are composed of coefficients in front of increasing powers of x.

So we can write a simple generalised expression for a power series as g of x,

equals a, plus bx,

plus cx squared, plus dx cubed et cetera.

Potentially, going off for

infinitely many times depending on what function we're considering.

When we calculate a Taylor series in the next video,

we will build up coefficient by coefficient,

where each term that we add improves the approximation.

In many cases, we will then be able to see a pattern emerge in the coefficients,

which thankfully saves us the trouble of calculating infinitely many terms.

However, many of the applications of Taylor series,

involve making use of just the first few terms of the series,

in the hope that this will be a good enough approximation for a certain application.

Starting from just a single term,

we call these expressions the zeroth,

first, second, and third order approximations etc.

Collectively, these short sections of the series are called Truncated series.

So let's begin by looking at some arbitrary,

but fairly complicated functions.

If we start with something like this,

where we have a function, let me just make it up.

Perhaps, shaped like this.

All we're going to do is focus on one particular point on this curve.

So let's say, this point here.

And then we're going to start building our function by trying to make it more,

and more like the point that we've chosen.

So, as the first term of our generalised power series is just a number a,

and we are ignoring all the other terms for now.

We know that our opening approximation must

just be a number that goes through the same point.

So we can just dive straight in,

and add our zeroth order approximation function to our plot.

It's going to be something like that.

Clearly, this hasn't done a great job with approximating the red curve.

So let's now go to our first order approximation.

This thing can also have a gradient.

And if we'd like to match our function at this point,

it should have the same gradient.

So we can have something a bit more like this.

Which is supposed to be a straight line.

Clearly, our approximation has improved a little in the region around our point.

Although, there is still plenty of room for improvement.

We can of course move onto the second order function,

which as we can see is a parabola.

Although at this point,

things get a little tough to draw.

And matching second derivatives by eye is also not easy.

But it might look something like this.

So we can say, okay let's go up and we're going to come down like that.

Hopefully, without having gone into any of the details about the maths,

you'll now be able to match up some mystery functions to

their corresponding truncated taylor series approximations in the following exercises.

In the next video, we're going to work through the detailed derivation of the terms.

But I hope this activity will help you not to

lose sight of what we're trying to achieve in the end.

See you then.