Ln(x) using repeated square root extraction

03212016, 05:09 AM
(This post was last modified: 03212016 05:12 AM by Gerson W. Barbosa.)
Post: #2




RE: Ln(x) using repeated square root extraction
For the sake of completeness here is the recursive limit definition of ln(x), even though this is not quite necessary for our purpose:
\[\ln (x)=\lim_{n\rightarrow \infty} \left [ n\cdot \left ( x^{\frac{1}{n}}1 \right )\sum_{k=2}^{\infty }\frac{\ln ^{k}(x)}{k!\cdot n^{k1}} \right ]\] If the limit is removed and n is set to 1 then we'll have the following recursive series representation: \[\ln (x)= \ x1 \sum_{k=2}^{\infty }\frac{\ln ^{k}(x)}{k!}\] 

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