Hey I have a test tommorow and am brushing up on some stuff...
Can someone please explain (not using too many complicated symbols =P)how to get the remainder for a Tailor series, ie using e(x)=1+x+(x^2)/2... as an example.
Thanks.
Hey I have a test tommorow and am brushing up on some stuff...
Can someone please explain (not using too many complicated symbols =P)how to get the remainder for a Tailor series, ie using e(x)=1+x+(x^2)/2... as an example.
Thanks.
I think you mean $\displaystyle e^x$, not e(x).
So I don't know exactly what you need but I'll try. Obviously a finite number of terms in a functions Taylor expansion is only an estimate and has an error, which is written in terms of of (n+1), with n terms in the expansion.
Generally, $\displaystyle R_{n+1}(x)=\frac{f^{n+1}(t)}{(n+1)!}(x-c)^{n+1}$, where c is the center and t is some value between x and a.
So for example let's say you are estimating $\displaystyle e^x$ centered at 0. Let's use 4 terms. As you said, the first four terms would be $\displaystyle 1+x+\frac{x^2}{2}+\frac{x^3}{6}$ and now let's say that the original function is these terms plus a remainder or error term.
$\displaystyle e^x =1+x+\frac{x^2}{2}+\frac{x^3}{6} + \frac{e^t}{24} x^4$ , where 0 < t < x.