I have a question: Why does this variation of Newton's iteration $x_{k+1} = x_{k} + \frac{f(x_{k})}{f'(x_k)}$, converge to a pole? My understanding of a pole is that it is the point at which the equation approaches infinity. I also believe that I need to expand the convergence with a Taylor series, but I am struggling on how to exactly prove WHY this formula converges to a pole. Any help is appreciated
2026-03-29 14:18:52.1774793932
Iteration converging to 0
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This iteration is Newton's method on the function $g(x) = 1/f(x)$, since $g(x)/g'(x) = -f(x)/f'(x)$.