Tylor Polynomials, Tylor Series and Power Series

Approximations and Taylor Polynomials

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

\[ e^x \cong 1 + x + \frac{x^2}{2} + \frac{x^3}{6} \]

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 \((a, f(a))\):

\[ y = f(a) + f'(a)(x - a) \]

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:

\[ P_N(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!} (x - a)^3 + \cdots + \frac{f^{(n)}(a)}{N!} (x - a)^N \]

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:

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