We have seen that we can approximate by its tangent
line to a function at a point locally derivable. If it holds that the function
is sufficiently soft at the point or study domain may be possible to try to
approximate a function by polynomials of degree one but also polynomials grade
two, three, four and so on. This approach is called the "Taylor polynomial
expansion" and is defined as follows:
P(x) =
f(a) + f’(a)(x-a) + f”(a)(x-a)2 +f”’(a)(x-a)3 + … + f n(a)(x-a)n
1!
2!
3!
n!
Where P (x) is the polynomial of degree n that best
approximates the function at the point x = a. Note that if we evaluate P (x) at
x = a all terms except f (a) are canceled, then P (a) = f (a). Note also that
the equation of the tangent line of the previous section corresponds to the
case in which n = 1.
Among all polynomials of order not greater than n,
that passing by f(a) the develop of Taylor’s polynomial expansion of f(x) on
x=a, is one that has the major order contact with f(x) en a. this is based on
the idea that if two functions share the same value at x=a, the same firs
derivative, the same second derivative an so on, (what we can express by saying
that both functions have the same contact of order i); so these functions would
be very similar rear to x=a, this means that we can approximate one function
for the other one committing a very insignificant error.
When a=0, this develop is called MacLaurin Series, In
practice most of the times we use MacLaurin series, and these are some examples.
ex = 1 + x + x2 + x3 + x4 + x5 + ….
1! 2! 3!
4! 5!
Sin(x) = x – x3 + x5 – x7 + …
3!
5! 7!
We must highlight that the symbol “=” in this case does not means equal, but an approximation, we have to add infinite terms to this expansion for this approximation become an equality.
This last step of adding infinite terms can not be taken lightly. We have said that the approximation of degree one, two, three etc is a local approximation to the point where the function is evaluated, ie, if we turn away a lot of the approximation point cease to be accurate. The more terms we add the Taylor series expansion of our approximation will be more accurate if we are in a neighborhood of the point. We might think that adding as infinite terms we can evaluate the approximate function at any point of their domain of definition with absolute precision. This is not always true, it depends on the character of the Taylor series at the point where we evaluate.
The character study of a series is a problem often complex. It is about defining the values for which the series is convergent, that is, determine the radius of convergence of it. Within the range of convergence of the series itself we can take infinite terms and admit that the series gives us the exact value of the function at the point. However, outside the range of convergence the series does not provide the exact value of the function though we add infinite terms.
The Taylor series expansions have great advantages when operating the functions whose equations involve complicated expressions, such as transcendental functions (sines, logarithms, etc..) However, it also has some drawbacks. A major drawback is that the number of terms necessary to approximate the function with reasonable accuracy at a point away from the evaluated (but always within the range of convergence of the series) is triggered to infinity. Another drawback is that the polynomial expression of the function can make it difficult to detect their basic properties, for example, it isn't obvious to infer that function sine is a periodic function.
