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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
3 years, 1 month 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.

- 3 years, 1 month ago