Integrating a Taylor series?



>>>>> "Alasdair" == Alasdair McAndrew <amca01 at gmail.com> writes:

    Alasdair> If I enter, say:

    Alasdair> define(t(x),taylor(f(x),x,a,10));

    Alasdair> where f(x) is a symbolic (that is, undefined) function,
    Alasdair> I get an expression full of derivatives, as I
    Alasdair> should. ?How can I integrate this series term by term? 
    Alasdair> ?Does Maxima have any flags or functions which will
    Alasdair> allow me to integrate a symbolic derivative?

As an example,

(%i10) taylor(f(x),x,a,1);
(%o10) f(a)+('at('diff(f(x),x,1),x = a))*(x-a)
(%i11) integrate(%,x,a,a+h);
(%o11) (h^2*('at('diff(f(x),x,1),x = a))+2*f(a)*h)/2

In 2D, which is easier to read:

                                     !
                         2  d        !
                        h  (-- (f(x))!     ) + 2 f(a) h
                            dx       !
                                     !x = a
(%o13)                  -------------------------------
                                       2

Isn't that the correct answer? What exactly were you expecting?

Ray