Does the sum $\frac{1}{3} + \frac{1}{3^{1+1/2}}+\cdots$ have a closed form?

Evaluate the sum $$\frac{1}{3} + \frac{1}{3^{1+\frac{1}{2}}}+\frac{1}{3^{1+\frac{1}{2}+\frac{1}{3}}}+\cdots$$

It seems that $1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}$ approaches $\ln n$ as $n\to \infty$, but I'm not sure if this is useful. Also, $3^{\ln n} =e^{\ln n\cdot \ln 3}= n^{\ln 3}$, but I'm also not sure how this is useful.

edit: I know how to prove that it converges, but I was wondering if there was a closed form for this sum.

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