Why is it not possible to generate an explicit formula for Newton's method?

Question

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.

Answer 1

Well, first of all, each application of Newton's method depends on the equation for which you wish to find roots.

Secondly, it doesn't always converge, i.e. there are conditions under which it does not converge.

Not knowing where you learned about Newton's method, I would suggest that you read further about it...try "Calculus" by M. Spivak.