Tylor Polynomials, Tylor Series and Power Series¶

Approximations and Taylor Polynomials¶

Here's a nice fact: for any real number x, we have:

e^{x} ≅ 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6}

Also, the closer x is to 0, the better the approximation.

Let's say we have some function $f$ which is very smooth, so that it can be repeatedly differentiated as many times as you like without causing any problems. Here is the question: what is the equation of the line which best approximates the curve $y = f (x)$ near the point $(a, f (a))$ ? The answer to this question is that the line we're looking for is the tangent line to the curve at the point $(a, f (a))$ :

y = f (a) + f^{'} (a) (x - a)

linear

Why stick to lines? What is the equation of the quadratic which best approximates the curve $y = f (x)$ near $(a, f (a))$ ? It turns out that the formula for the quadratic which best approximates the curve $y = f (x)$ for $x$ near $a$ is given by:

y = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2} (x - a)^{2}

This is actually a quadratic in $x$ , because if you expand $(x - a)^{2}$ , the highest power of $x$ floating around is $x^{2}$

Let's call the quadratic $P_{2}$ :

P_{2} (x) = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2} (x - a)^{2}

Let's continue the same pattern, except that we'll use some arbitrary degree $N$ instead of just 1 or 2. So, here's our question: which polynomial of degree $N$ or less gives the best approximation to $f (x)$ for values of $x$ near $a$ ? The answer is given by A Tylor approximation theorem:

If $f$ is smooth at $x = a$ , then of all the polynomials of degree $N$ or less, the one which best approximates $f (x)$ for $x$ near $a$ is given by:

P_{N} (x) = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2!} (x - a)^{2} + \frac{f^{(3)} (a)}{3!} (x - a)^{3} + \dots + \frac{f^{(n)} (a)}{N!} (x - a)^{N}

In sigma notation, the formula looks like this:

P_{N} (x) = \sum_{n = 0}^{N} \frac{f^{(n)} (a)}{n!} (x - a)^{n}

We call the polynomial $P_{N}$ the Nth-order Taylor polynomial of $f(x)$

Now we have a approximation for $f (x)$ , but how good is it? Here comes another formula for the error, the Taylor' Theorem:

In above contents we have a function's Nth-order Taylor polynomial about $x = a$ :

P_{N} (x) = \sum_{n = 0}^{N} \frac{f^{(n)} (a)}{n!} (x - a)^{n}

We want to use the value of $P_{N} (x)$ to approximate the true value $f (x)$ , so we consider the error term, which is the difference between the true value and the approximate value:

R_{N} (x) = f (x) - P_{N} (x)

Actually, $R_{N} (x)$ is called the Nth-order term, it's also referred to as the Nth-order remainder term. Tylor's Theorem gives an alternative formula for $R_{N} (x)$ :

R_{N} (x) = \frac{f^{(N + 1)} (c)}{(N + 1)!} (x - a)^{N + 1}

where $c$ is some number which lies between $x$ and $a$ .

Since $f (x) = P_{N} (x) + R_{N} (x)$ , we can write:

f (x) = \sum_{n = 0}^{N} \frac{f^{(n)} (a)}{n!} (x - a)^{n} + \frac{f^{(N + 1)} (c)}{(N + 1)!} (x - a)^{N + 1}