I recently came across the following infinite series on the website "Art of problem solving" (which is more like a discussion forum):. https://artofproblemsolving.com/community/c7h1807181p12023116
$$\sum_{n=1}^{\infty} \bigg(\bigg(\frac {1}{3} \bigg)^{\sum_{k=1}^{n} \frac {1}{k} } \bigg) = \bigg( \frac {1}{3} \bigg) + \bigg( \frac {1}{3} \bigg)^ { 1+ \frac {1}{2} } + \bigg( \frac {1}{3} \bigg)^{1+ \frac {1}{2} + \frac {1}{3}} + ...$$ Where a user proved that the series is convergent and Wolfram Alpha shows Its value to be around $5.3$.
So, I tried to find the closed form for this series but couldn't really get anything. I want to know if the answer would even be in terms of elementary functions or not, since I don't want to waste a lot of time on something that is not possible for me.
If this has a known closed form, or if anyone can solve for one, please let me know if it's made up of ONLY elementary functions (like polynomials, logarithms, exponential functions, etc). (Also, you may suggest any changes that you might want in how this question is framed) Thanks!