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Nested Radical Illusion

Consider the infinite radical \[1+\sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\sqrt[4]{\frac{1}{4}+\sqrt[5]{\frac{1}{5}+...}}}}.\] then the infinte radical can be less than a approximation of \( \sqrt[b]{a\pi} \), find \(a+b\) ?

where a and b are integer positive.


How tight can we make this bound?

Note by Uzumaki Nagato Tenshou Uzumaki
2 years, 10 months ago

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Calculating the radical out to the 14th root gives a value of \(2.272225\) to 6 decimal places, and there isn't significant variation from this value from the 10th root onward. So taking this as a "target value", we can get arbitrarily close by choosing larger and larger values of \(a\) and \(b\). For example, we have that \(\sqrt[9]{514\pi} = 2.272255\) to 6 decimal places, giving a difference of \(0.0013\)%. Even closer is \(\sqrt[15]{70733\pi} = 2.272226\) to 6 decimal places.

So we can make the bound arbitrarily tight, but perhaps the more interesting question is whether we can find an exact value with \(a,b\) positive integers. It's unlikely, as such a beautiful result would probably be well-known and have a name attached to it, but since it's virtually impossible to prove that there isn't an exact solution there is still a chance.

Brian Charlesworth - 2 years, 10 months ago

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