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A function f has a root of multiplicity $m>1$ at the point $ x_*$ if $f(x_*)=f'(x_*)=...=f^{(m-1)}(x_*)=0$. Assume that the iteration$ x_{k+1}=x_k-mf(x_k)/f'(x_k)$ converges to $x_*$. If$ f^{(m)}(x_*)≠0$, prove that this sequence converges quadratically.

(We may use the Taylor's series, but I cannot get the result we need to prove.

Expand expand $f(x_k)$ around $x_*$ until $m$-th order derivative term, which has the form

$(x_k - x_*)^m f^{(m)} (x_k) / m!$, and similarly for $ f ' (x_k)$ )

(We may use the Taylor's series, but I cannot get the result we need to prove.

Expand expand $f(x_k)$ around $x_*$ until $m$-th order derivative term, which has the form

$(x_k - x_*)^m f^{(m)} (x_k) / m!$, and similarly for $ f ' (x_k)$ )

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