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?

#Algebra #Calculus #NestedRadicals

Easy Math Editor

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Comments

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 - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$