DinS Written on 2018/4/30

1. A brief introduction to power series

Before we jump into Taylor series, let’s first look at something called power series.

If a series looks like this format, it’s a power series. We’ve already seen some. For example, when a_n =1/n, x = 1, it’s a harmonic series. When a_n= 1, x = 1/2, it’s a geometric series with r = 1/2.

Now look at a more general one. If a_n = n, x = 3, it becomes like this.

If we write the series out it will look like this.

This somehow looks like a polynomial function. Indeed. There’s something we can do between these two.

Clearly not all power series converge. The above is an example. It depends on x.

We’ll define a term called interval of convergence. It is a set for all real numbers x so that the series converge. From here on we’ll assume that all x falls in interval of convergence, which simply means that the power series will converge and we can write a function about it.

We can do something interesting about power series.

We can differentiate it.

That is we’re taking a_n as a constant and apply power rule to x.

We can also integrate it.

One application is that we can use this trick to find function for other series from a given series.

We’ve already know that

That’s what we’ve got from geometric series.

Now what if we integrate it?

Pardon me for being sloppy here. I just want to show you the whole picture, the idea.

From this we can get

By this way we’ve found the function for a new series.

2. Taylor Series: motivation

Up to now, all we’ve been doing is starting from certain series to get its function representation, if it exists.

Here we’ll start from certain function to get its series. That’s basically what Taylor series is meant to do.

Let’s first look at this problem.

From previous articles we know linear approximation.

f(x) ≈ f(a) + f’(a)*(x-a)

We can do better by using polynomial approximation.

We’ve added a secondary derivative to linear approximation. What’s the meaning of g(x)?

Actually g(a) = f(a), clearly.

This means that g’(a) = f’(a).

This means that g’’(a) = f’’(a).

All this leads to one thing: we’ve constructed a function g(x) that behaves likes f(x).

We can still do better by adding higher derivates.

Let’s see an example.

f(x) = sinx. Suppose we have worked out the derivatives of trig-functions. We have

This is a much better approximation to sinx.

Surely beats linear approximation.

We can still do better, but this time we’re not going to add more terms. Although more terms mean better approximation, it’s finite. We know power series share some similarity with polynomial function. Maybe we can use power series with infinite terms to do such thing.