PROOF OF TAYLOR'S THEOREM
Proof: Taylor's theorem in one variable
 Integral version
We first prove Taylor's theorem with the integral remainder term.
The fundamental theorem of calculus states that
which can be rearranged to:
Now we can see that an application of Integration by parts yields:
The first equation is arrived at by letting and dv = dt; the second equation by noting that ; the third just factors out some common terms.
Another application yields:
By repeating this process, we may derive Taylor's theorem for higher values of n.
This can be formalized by applying the technique of induction. So, suppose that Taylor's theorem holds for a particular n, that is, suppose that
We can rewrite the integral using integration by parts. An antiderivative of (x − t)n as a function of t is given by −(x−t)n+1 / (n + 1), so
Substituting this in (*) proves Taylor's theorem for n + 1, and hence for all nonnegative integers n.
The remainder term in the Lagrange form can be derived by the mean value theorem in the following way:
where ξ is some number from the interval [a, x]. The last integral can be solved immediately, which leads to
More generally, for any function G(t), the mean value theorem asserts the existence of ξ in the interval [a, x] satisfying
Rn is the remainder :
SOURCE : WIKIPEDIA