polynomial approximates function

[kostas1173]

Hi, I hope I'm in the right place to ask.
What does it mean that a polynomial Pn(x)=1+x+x^2/2!+..+x^n/n! approximates the function f(x)=e^x?

Thank you

 

[tkhunny]

What does it mean to write:

\(\displaystyle P_{n}(x)\)

???

 

[kostas1173]

I think it means that the polynomial depends on n(denominator) and x.

 

[galactus]

It is the Taylor series for e. the more terms you add, the more accurate it becomes.

More specifically, being centered at 0, we call it the MacLaurin series.

Say you want \(\displaystyle e^{1}\). Sub x=1 into the Taylor expansion for e.

\(\displaystyle 1+1+\frac{1^{2}}{2}+\frac{{1}^{3}}{6}+\frac{{1}^{4}}{24}+\frac{1^{5}}{5!}+\frac{1^{6}}{6!}=2.71805555556\)

The more terms you add, the better the approximation. This is pretty close so far because \(\displaystyle e^{1}=2.71828182846\)

 

[kostas1173]

Thank you very much. It makes sense now.