I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the remainder exists but not the mathematical description. Why is the value being plugged into the derivative of the remainder some number between $x$ and $a$? What is the connection to the mean value theorem? Is the remainder used to bound the function? Lastly and my most important question is what is the intuitive (for me most likely a geometrical approach would be great) for how this remainder is derived. I've spent a long time trying to figure it out for myself and looking online but it seems I'm missing something for there are virtually no questions being asked about this.
Thanks for you time, Jackson