Infinite Root Integral

The problem can be rewritten as:

$\displaystyle\int_{0}^{1}(x^{1/2}x^{1/6}x^{1/24}...)dx$

= $\displaystyle\int_{0}^{1}(x^{1/2+1/6+1/24+...})dx$

We then know that $e = 1 + 1 + 1/2 + 1/6 + 1/24 + ...$ , therefore we have

$\displaystyle\int_{0}^{1}(x^{e-2})dx$

$= (x^{e-1})/(e-1)$

And we get our answer of $\frac{1}{e-1}$ , or 0.5819