Square Root Infinity?

Calculus Level 1

Take any positive number and find its square root, which you call $a.$

Now, find the square root of $a$ and call the value $b.$

Then find the square root of $b$ and call the value $c,$ and so on.

If you did this indefinitely, what number would the limiting value converge to?

1 3 2 0

1 solution

Alex G
Apr 20, 2016

From the problem we get:

$\sqrt{\sqrt{...\sqrt{a}}}$

Which is equivalent to:

$\lim_{x\to\infty}{a^{\frac{1}{2^x}}}$

Let $L$ equal the limit, and take the natural log of both sides

$\ln{L}=\lim_{x\to\infty}{\frac{1}{2^x}×\ln{a}}$

$\ln{L}=0$

$L=\boxed{1}$

I remember doing this in calculators just for fun when I was 10. Nice question @Pi Han Goh

Aditya Kumar - 5 years, 1 month ago