Tonight I was messing with the integration by parts formula and found the following formula. Starting with the usual $$\int udv = uv - \int vdu$$ and letting $u = f(x)$, $du = f'(x)dx$, $dv =dx$ and $v=x$ we obtain $$\int f(x)dx = xf(x) - \int xf'(x).$$ We then apply the apply the integration by parts formula again to the integral on the RHS to obtain $$\int f(x)dx = xf(x) - \frac{f'(x)x^2}{2} + \frac{1}{2}\int x^2f''(x)dx$$ and again: $$\int f(x)dx = xf(x) - \frac{f'(x)x^2}{2} + \frac{f''(x)x^3}{2\cdot 3} - \frac{1}{2\cdot 3}\int x^3f'''(x)dx$$ So, in general, it seems (at least to me) safe to say that $$\int f(x) dx = xf(x) - \frac{f'(x)x^2}{2} + \frac{f''(x)x^3}{2\cdot 3} - \frac{f'''(x)x^4}{2\cdot 3\cdot 4} + \ldots =\sum_{n=1}^{\infty}\frac{x^nf^{(n)}(x)}{n!}(-1)^{n+1}$$ Next, applying the alternating series estimation theorem, we have that $$|E_N|=\left|\left(\sum_{n=1}^{\infty}\frac{x^nf^{(n)}(x)}{n!}(-1)^{n+1}\right) - \left(\sum_{n=1}^{N}\frac{x^nf^{(n)}(x)}{n!}(-1)^{n+1}\right)\right| \leq \frac{x^{(N+1)}f^{(N+1)}(x)}{(N+1)!}(-1)^{N+1}$$ I then tried to use this estimate to approximate $\int_{a}^{b}f(x)dx$. That is, since we have $$E_N \geq -\frac{x^{(N+1)}f^{(N+1)}(x)}{(N+1)!}(-1)^{N+1}$$ we can conclude that $$\int_{a}^{b}f(x)dx \approx \left(\sum_{n=1}^{N}\frac{x^nf^{(n)}(x)}{n!}(-1)^{n+1}\right)-\frac{x^{(N+1)}f^{(N+1)}(x)}{(N+1)!}(-1)^{N+1}.$$
It then occured to me that to use the alternating series estimation theorem we must have that $$0\leq\frac{x^{n+1}f^{(n+1)}(x)}{(n+1)!}\leq \frac{x^{n}f^{(n)}(x)}{(n)!}$$ and $$\lim_{n \to \infty}\frac{x^nf^{(n)}(x)}{n!} = 0$$ So it seems that the above approximation only works if we choose an $f(x)$ so that the above hypothesis are satisfied. Is there a way to estimate the infinite sum that does not rely on the alternating series test so we do not have to worry about satisfying the above restrictions?
More generally, I am interested if there is a more interesting route to take with this process of repeated integration by parts. The form of the terms in the infinite series remind me of taylor series. Is there a connection?