Consider the sequence
[tex]a_1=2[/tex]
[tex]a_2=2\sqrt2[/tex]
[tex]a_3=2\sqrt{2\sqrt[3]{2}}[/tex]
[tex]a_4=2\sqrt{2\sqrt[3]{2\sqrt[4]{2}}}[/tex]
and so on. We can simplify each of these terms as
[tex]a_1=2^1[/tex]
[tex]a_2=2^{1+1/2}[/tex]
[tex]a_3=2^{1+1/2+1/6}[/tex]
[tex]a_4=2^{1+1/2+1/6+1/24}[/tex]
and so on, with the general pattern
[tex]a_n=2^{1+1/2+1/6+\cdots+1/n!}[/tex]
As [tex]n\to\infty[/tex], the exponent takes on the value of [tex]e-1[/tex], since
[tex]e=\displaystyle\sum_{n\ge0}\frac1{n!}[/tex]
Then the sequence [tex]a_n[/tex] converges to [tex]2^{e-1}\approx3.29044[/tex].
