Going through the recursive formula for approximating roots every time is extraordinarily tedious, so I was wondering why there was no formula that computed the $n$th iteration of Newton's method.
Calculus – Why No Explicit Formula for Newton’s Method?
calculusnewton raphsonnumerical methods
Best Answer
First, note that such a formula would violate the "too good to be true" test: it would let you find the minima of any (sufficiently smooth, strictly) convex function in a single step, no matter how complicated!
For intuition, consider what Newton's method is doing: you are starting from an initial guess, and taking a sequence of "downhill steps" until you reach the minimum, where each step uses local information (a quadratic local approximation) about your neighborhood to compute the next destination. Imagine a hiker climbing down a mountain, who first hikes to the lowest point she can see, then reorients herself and goes to the lowest point she can see from her new location, etc.
The only way to make a direct beeline to the lowest point ($n\to\infty$) is if the hiker somehow knows the entire shape of the landscape in advance. In mathematical terms, this means your formula would have to use either:
And of course neither is practical in practice. Note though that sometimes you do have such complete information, e.g. when your function is quadratic, so that all higher-order derivatives are zero. And then you can easily make a beeline straight to the solution.
Also, you can write down a formula for the $n$th iteration, but as stated in the other answers, this is not useful: it gets messier and messier the larger $n$ gets (as it must, as explained above), doesn't converge to anything nice as $n\to \infty$, and amounts to nothing more than... running Newton's method for $n$ consecutive steps.