Question

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.

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`.