Is there a closed form solution for partial sums of $1/(2^{2^0}) + 1/(2^{2^1}) + 1/(2^{2^2}) + \ldots$ Title says it all, this is such a classical looking series,
$$\frac1{2^{2^0}} + \frac1{2^{2^1}} + \frac1{2^{2^2}} + \ldots.$$
So, I was just wondering, is there a closed form solution known for the partial sums?  If so, can someone post?  I've looked around a bit and haven't found a closed form solution yet.

I doubt there's a closed expression for it, but the partial sums arise here:
http://oeis.org/A085010
and in particular the decimal expansion of the infinite series is here:
http://oeis.org/A007404
There is a (paywall) article called "Simple continued fractions for some numbers" where the problem being considered is to find the continued fraction expansion of $\sum_{n=0}^\infty \frac{1}{U^{2^n}}$. It doesn't seem to be known how to find a closed form expression for the sum however.
