A Taylor Series is a polynomial function with an infinite number of terms, expressed as an Infinite Series. Taylor Series can be used to represent any function, as long as it is an analytic function. If the function is not infinitely differentiable, Taylor Series can be used to approximate values of a function. Either way, the approximation will be more accurate along a certain interval of convergence.
Taylor Series Basics
To understand Taylor Series, let's first construct a polynomial: P(x) = a0 + a1x + a2x2 + a3x3 + a4x4 + ... + anxn + ...
or, in other words
\ a x
But what do we use to represent an
? First, lets take a step back and investigate the derivatives of this polynomial. Once you take a few derivatives, you'll find that the following pattern appears: the coefficient of P( n )(x) is an * n!
With this knowledge, we can now...
Construct a Basic Taylor Series
Perhaps the most basic Taylor Series is ƒ(x) = ex. We'll use this function to derive our first Taylor Series, centered at x = 0. Our objective is to make the polynomial we constructed above resemble this function. But how are we going to do this? Easy! Take the derivatives of both P(x) and ƒ(x) and set them equal.
First, we must start "cranking out" derivatives. This is easy, as all the derivatives of ƒ(x) = ex are ex!
ƒ(x) = ex ƒ(0) = 1
ƒ'(x) = ex ƒ'(0) = 1
ƒ''(x) = ex ƒ''(0) = 1
ƒ'''(x) = ex ƒ'''(0) = 1
ƒIV(x) = ex ƒIV(0) = 1
etc, etc, ad nauseum
Now, we can set P(0) = ƒ(0), P'(0) = ƒ'(0), P''(0) = ƒ''(0), and so forth. But this comes with a catch. Remember how above we came up with the formula for P( n )(x) is an * n! ? This means we must divide the derivatives of ƒ(x) by n!. Thus, we get the general formula for a Taylor series centered at x = 0:
--- n n
\ ƒ (0) x
Congratulations! You just constructed your first Taylor Series for ƒ(x) = ex, centered at x = 0. Since all of its derivatives at 0 are 1, the sigma notation for this series is:
If you graph this, you will see that the polynomial curve starts to fit to the graph of ex, and fits even better as you add more terms to the polynomial. Basically, this is how your calculator preforms advanced operations (integrals, etc.) on complex functions, because polynomials are much easier to work with compared to the complex function you may provide.
We just created a special type of Taylor Series, because we chose to center our approximation at x = 0. This type of series is specifically known as a MacLaurin Series, named for the mathematician who discovered it. The general formula for a Taylor Series centered at x = a is:
--- n n
\ ƒ (a) (x-a)
Constructing other Taylor Series from known Taylor series
Now that we know the Taylor Series for ƒ(x) = ex centered at x = 0, let's construct a series for g(x) = ex-1, centered at x = 0.
This is quite easily done. All we have to do is take the original Taylor Series for ƒ(x) and subtract 1! Now say we wanted to construct a Taylor Series for h(x) = ex-1 / x, centered at x = 0. All we have to do for this is take the series for g(x) and divide by x! Both of these can still be easily represented with sigma notation:
g(x)= \ x
h(x)= \ x
is valid for all algebraic
operations and especially useful for derivative
s and integral
s. For more information, read on!
Common Taylor Series Useful in Forming Other More Complex Series
Nota Bene: All these series are centered at x = 0.
S(x) = sin(x)
S(x) = \ n x
/ (-1) ------
C(x) = cos(x)
C(x) = \ n x
/ (-1) -----
L(x) = 1/(1+x)
--- n n
L(x) = \ (-1) x
See also PMDBoi's write-up above for more useful Series.
Delving a Bit Deeper: Intervals of Convergence
Once you start using these Taylor Series, you will start to notice that some will fit the curve of the actual function better as you increase the number of polynomial terms, others will fit better until a point, and then there are even some that only fit the curve when they equal the point chosen to center the series around. To further understand this, we must analyze the interval of convergence.
Theorem: Interval of Convergence
P(x)= \ a z
be a power series. Then there is an extended real number R ( 0 ≤ R ≤ ±∞) such that:
1.) P(x) converges for all z such that z is a subset of C and |z| < R
2.) P(x) diverges for all z such that z is a subset of C and |z| > R
Now that we have that established, just how do we find that number R? Going back to our basic knowledge of mathematical series, we have a veritable cornucopia of options for testing for convergence. Some of the best for power series are the alternating series test (useful when the terms alternate signs, like in sin(x) and cos(x)), and there's always the good Ratio Test. The Ratio Test is usually the best choice for Taylor Series because they contain exponentials and/or factorials. Let's now find the interval of convergence for f(x) = ex using the Ratio Test.
| n+1 |
lim | x n! |
n→∞ | ----- * ---- | < 1
| (n+1)! n |
| x |
lim | x |
n→∞ | ----- | < 1
| (n+1) |
Thus, we get 0 ≤ 1. Since this is ALWAYS true, our R is ∞! This case means that the fit of the Taylor Polynomial will increasingly get better as more polynomial terms are added. As the number of terms approaches infinity, the EXACT function will appear!
There are two other cases that arise:
A.) The interval of convergence is |x| ≤ k. This is the case mentioned earlier where an increasing number of polynomial terms can be added, but a point is reached where adding more terms does not make a better fit. This is because the series diverges from the actual graph of the function around x = k and x = k.
B.) The interval of convergence is a. This is the last case mentioned above where the actual graph of the function only matches the Taylor Series Polynomial where the two graphs intersect. This usually occurs at the point where the Taylor Polynomial was centered, x = a. This interval of convergence is only good for calculating the value of the function at that specified point.