How do we find more appropriate constants for expansions of functions? We all knonw that the expansion of  $e^x$ is $$1+x+x^2/2+...$$.
But what if I want to find more approximate expansion of $e^x$. I try that
$$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and substituting the values $x=1,2,3$ to get these constants. And I get more approximate results by solving 3 equations. Is there any way to find more appropriate constants for this function.Which values of $x$ must be taken to get less error?
I appreciate any help.
 A: The usual power series
fits a function at a point
(often zero)
and gets worse
the further you get
from that point.
To get an approximation
that is good over a 
whole interval,
look at Chebychev approximations
or least squares fitting.
A: you are approximating a function using polynomial. If this is a not homework question, you should find a textbook on approximation using spline function since what you have done is close to B-spline.
A: The Taylor series around a certain point $x_0$ is already an exact expansion. Usually what you use is the McLaurin series, which is the taylor series for $x_0 = 0$. Thence you cannot fine a "more approximate" expansion of $e^x$. What you can do is to write down the general expansion of $e^x$ for a point $x_0 \neq 0$, which would read:
$$e^x = e^{x_0}\left(1 + (x - x_0) + \frac{1}{2}(x - x_0)^2 + \frac{1}{3!}(x - x_0)^3 + \cdots\right) = e^{x_0} \sum_{k = 0}^{+\infty} \frac{(x - x_0)^k}{k!}$$
The coefficients $1, \frac{1}{2}, \frac{1}{6}, \cdots $ arise naturally by the definition of Taylor Series itself, and they will be different for different functions.
