Family of 3's

$\begin{aligned} x & = \sqrt[4]{3 \sqrt[7]{3^2\sqrt[10]{3^3\sqrt[13]{\cdots}}}} \\ & = \left(3 \left(3^2 \left(3^3 \left(\cdots \right)^\frac 1{13} \right)^\frac 1{10} \right)^\frac 17 \right)^\frac 14 \\ & = 3^\frac 14 \cdot 3^{2 \cdot \frac 14 \cdot \frac 17} \cdot 3^{3 \cdot \frac 14 \cdot \frac 17 \cdot \frac 1{10}} \cdots \\ & = \exp \left(\ln 3 {\color{#3D99F6} \left(\frac 14 + 2 \cdot \frac 14 \cdot \frac 17 + 3 \cdot \frac 14 \cdot \frac 17 \cdot \frac 1{10} + \cdots \right)} \right) & \small \color{#3D99F6} \text{where }\exp (x) = e^x. \\ & = \exp \left({\color{#3D99F6}\frac 13} \ln 3\right) & \small \color{#3D99F6} \text{See note.} \\ & = \sqrt[3] 3 \approx \boxed{1.44} \end{aligned}$

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Note: See the proof of $\displaystyle \sum_{n=1}^\infty \frac n{\prod_{k=1}^n (3k+1)} = \frac 13$

View A Nested Radical . It has got a full discussion on how this nested radical and some of its friends are formed as well as how an infinite series from this is concluded.