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This appeared in the Area Level of the 19th Philippine Mathematical Olympiad. Electronic calculators were not allowed during the competition of course. The closest I got to was to express it as:

$$ \dfrac{1}{\sqrt[6]{2}(\sqrt[6]{7813}-\sqrt[3]{6}\sqrt[6]{217})}. $$

The answer using an electronic calculator is 9375. Can anyone show how this can be approximated without the use of electronic calculators?

StubbornAtom
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hpesoj626
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1 Answers1

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Consider $$y=\frac{1}{\sqrt[6]{x^6+1}-\sqrt[6]{x^6-1}}$$ and use series expansions for "large" values of $x$ for each piece and then long division.

You should end with $$y=3 x^5-\frac{55}{72 x^7}+O\left(\frac{1}{x^{19}}\right)$$

Claude Leibovici
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    In this case, as we're interested in estimates at a specific input value, I think the answer is not technically complete without estimates / upper bounds for the coefficient of any of the truncated terms. One cannot tell directly from what you've written here that the $x^{-19}$ term doesn't have a coefficient of size $5^{19}$ (or bigger). And what about the $x^{-31}$ term and so on? Of course, you're free to leave such details out, but I think it at least ought to be mentioned. – Arthur Jun 19 '19 at 07:01