I have read through many of the (many) posts related to Taylor's Inequality / Remainder; however, I wondered if there were an intuitive way of understanding the error from point-of-view of series / convergence?
If I have understood correctly, for a function $f(x)$, the error on the $n^{th}$-order Taylor series $T_n(x) = \sum_{i=0}^{n} \frac{f^{(i)}(c )}{i!}(x-c)^n$ is given exactly by either the integral-form or Lagrange remainder:
$$ R_n(x) \; = \; f(x) -T_n(x) = \frac{1}{n!} \int_c^x (x-t)^n f^{(n+1)}(t) \;\mathrm{d}t % \; = \; \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}, $$ for some $z \in [x,c]$.$^{\color{red}1}$
I have managed to replicate the proof that use the FTC and induction, and have read through at least three others, so I have seen and "know" where these equivalences come from; however, they don't "make sense" to me from a either a graphical or series (i.e. convergence) perspective.
In other words, if we express the remainder as the tail-end of the complete Taylor series for $f(x)$, is there any way we might (intuitively) anticipate or expect:
$$ \sum_{i=n+1}^{\infty} \frac{f^{(i)}(c )}{i!}(x-c)^n \leq \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}, $$
i.e. that this series would be bounded in this way / that it would converge to a value that can be given in such a particular form (specifically a value closely related, in form, to the first term of the series, i.e. $R_n(x)$)? I'm trying to express the question in terms of series, as clearly as possible, but I'm not 100% certain of the correct way of doing so.
Especially in any case where the terms of the remainder were all positive, it seems so incredibly unintuitive / unexpected that the sum on the left would be bounded by the value on the right (let alone converging to it).
Similarly, I have no idea how to begin to think about the integral form, in graphical terms. Because it's stated as an integral (& because $f(x)-R_n(x)$ has such an obvious graphical interpretation), I feel like there should be some clear graphical interpretation, but I've not the faintest clue of where to start (I'm not sure if this constitutes a separate question; can post separately, if need be).
This is frustrating, because I feel like I really follow the proofs, and I can apply the result to standard problems from an applied-maths course, but no matter how many times I come back to it, or how much I read, it's not the least bit intuitive, from either of these perspectives.
Thanks in advance for your time and any help / references!
$^{\color{red}1}$: N.B. I'm following the terminology used in a textbook; apologies if it is not standardized. "Integral form" seems an awkward title and the presentation does not suggest that it is "the integral form of the Lagrange remainder," which is what I would have otherwise guessed.