Find the value of 2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\sqrt[6]{2\sqrt[7]{...}}}}}} An index is going to infinity.

Question

asked Feb 11, 2020 163k views

1 Answer

Dialer · Answer 1 · 2020-02-15T09:13:23+0000

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
$2^(e-1)\approx3.29044$ .