Question

Find the value of 6%5D%7B2%5Csqrt%5B7%5D%7B...%7D%7D%7D%7D%7D%7D" id="TexFormula1" title="2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2\sqrt[7]{...}}}}}}" alt="2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2\sqrt[7]{...}}}}}}" align="absmiddle" class="latex-formula">
An index is going to infinity.

Answer 1

Consider the sequence

$a_1=2$

$a_2=2\sqrt2$

$a_3=2\sqrt{2\sqrt[3]{2}}$

$a_4=2\sqrt{2\sqrt[3]{2\sqrt[4]{2}}}$

and so on. We can simplify each of these terms as

$a_1=2^1$

$a_2=2^{1+1/2}$

$a_3=2^{1+1/2+1/6}$

$a_4=2^{1+1/2+1/6+1/24}$

and so on, with the general pattern

$a_n=2^{1+1/2+1/6+\cdots+1/n!}$

As $n\to\infty$, the exponent takes on the value of $e-1$, since

$e=\displaystyle\sum_{n\ge0}\frac1{n!}$

Then the sequence $a_n$ converges to $2^{e-1}\approx3.29044$.