I have been playing around with the fundamental theorem of calculus:
Starting with $$f(x)=f(x_0)+\int_{x_0}^x f'(t_1) \mathrm{d}t_1, \tag{1}$$
one can apply the FTC again, this time to the derivative, to get
$$f'(t_1)=f'(x_0)+\int_{x_0}^{t_1} f''(t_2) \mathrm{d}t_2. \tag{2}$$
Substituting $(2)$ in $(1)$ gives
$$f(x)=f(x_0) + f'(x_0) (x-x_0)+\int_{x_0}^x \int_{x_0}^{t_1} f''(t_2) \mathrm{d}t_2\, \mathrm{d}t_1.$$
This is one way to get the classical Taylor expansion, with the remainder term here in iterated integral form.
I wondered what would happen if different "centers" are used at every step. For example, replacing Equation $(2)$ with $$f'(t_1) = f'(x_1)+ \int_{x_1}^{t_1} f''(t_2) \mathrm{d}t_2, $$ gives
$$f(x)=f(x_0)+f'(x_1)(x-x_0)+\int_{x_0}^x \int_{x_1}^{t_1} f''(t_2) \mathrm{d}t_2 \, \mathrm{d} t_1. $$
What I got after $n$ steps was
$$f(x)=P_n(x;x_0,x_1,\dots,x_n)+R_n(x;x_0,x_1,\dots,x_n),$$ where
$$P_n(x;x_0,x_1,\dots,x_{n})=\sum_{k=0}^n f^{(k)}(x_k) \int_{x_0}^x \int_{x_1}^{t_1} \cdots \int_{x_{k-1}}^{t_{k-1}} \mathrm{d}t_k \, \cdots \mathrm{d}t_2 \, \mathrm{d} t_1, $$
and
$$R_n(x;x_0,x_1,\dots,x_{n}) = \int_{x_0}^x \int_{x_1}^{t_1} \cdots \int_{x_{n-1}}^{t_{n-1}} \int_{x_n}^{t_n} f^{(n+1)}(t_{n+1}) \mathrm{d}t_{n+1}\,\mathrm{d}t_n \, \cdots \mathrm{d}t_2 \, \mathrm{d} t_1.$$
Some of the low-order polynomials are:
$$\begin{align} P_0 &= f(x_0), \\ P_1 &= f(x_0)+f'(x_1)(x-x_0), \\ P_2 &= f(x_0)+f'(x_1)(x-x_0)+ \frac{f''(x_2)}{2}(x - x_0) (x + x_0 - 2 x_1),\\ P_3 &= P_2+ \frac{f'''(x_3)}{6} (x-x_0) \left(x^2-3 x_1^2+x_0 \left(x+x_0\right)-3 \left(x+x_0-2 x_1\right) x_2 \right). \end{align} $$
My questions are:
- Do the multivariate polynomials $\int_{x_0}^x \int_{x_1}^{t_1} \cdots \int_{x_{k-1}}^{t_{k-1}} \mathrm{d}t_k \, \cdots \mathrm{d}t_2 \, \mathrm{d} t_1$ have a closed form?
- Is this expansion well-known?
- Aside from polynomials, and analytic functions where $x_0=x_1=\dots$ , are there criteria for which an infinite series expansion exists? (that is $\lim_{n\to \infty} R_n =0 $.)
Thanks.