Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Question

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding equivalent of function using polynomials or other method is approximation? How do we know the Taylor series for a function is approximation and not exact? Also, not necessarily, I want to know is there some way of finding exact equivalent of function? And why not?

Answer 1

If it comes to computing concrete function values as decimals, at the very bottom the only operations you have at your disposal are the famous four: $+$, $-$, $\times$, $:\>$. So it seems that the only functions you can compute exactly in a finite number of steps are rational functions (or piecewise concatenations of such).

But we all know that there are very interesting functions out there which are not of this kind: $\exp$, $\log$, $\cos$, etcetera. These functions are defined by "inner properties", like $f(x+y)=f(x)\cdot f(y)$, or $y''+y=0$. In order to use them in our daily dealings with geometrical or physical matters we need procedures that find the concrete values of these functions with prescribed accuracy. One such procedure is the use of Taylor series, and there are many others.

Answer 2

For about the same reason that you don't write the decimal expansion of rational numbers in full: that takes an infinite number of digits.

The so-called transcendent functions usually take an infinite number of arithmetic operations to be evaluated exactly, whatever the approach.

And functions are also often defined from approximations that can be refined at will by increasing the number of terms (we say that these approximations converge to the function).

For instance, the series

$$1+x+\frac{x^2}2+\frac{x^3}{2\cdot3}+\frac{x^4}{2\cdot3\cdot4}+\cdots$$ is a way to define the exponential function $e^x$, if you consider an infinity of terms.

A secondary argument is that mathematicians often need to discuss the properties of the functions by replacing them with similar ones for which suitable properties are already known.

For the sake of the example, the exponential can be bounded by a crude linear approximation,

$$e^x\ge 1+x$$ and this is enough to prove that the value of the exponential can be as large as you want.