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Let $S$ be the given expression. Consider the binomial expansion below.

$\begin{aligned} (1+x)^\frac 32 & = 1 + \frac {3x}{2^11!} + \frac {3 x^2}{2^22!} - \frac {3x^3}{2^33!} + \frac {3\cdot 3 x^4}{2^4 4!} - \frac {3\cdot 3 \cdot 5 x^5}{2^5 5!} + \cdots \\ & = 1 + \frac {3x}{2^11!} + \frac {3 x^2}{2^22!} - 3\left(\frac {x^3}{2^33!} - \frac {1\cdot 3 x^4}{2^4 4!} + \frac {1\cdot 3 \cdot 5 x^5}{2^5 5!} + \cdots \right) & \small \color{#3D99F6} \text{Putting }x = 1 \\ 2^\frac 32 & = 1 + \frac {3}{2^11!} + \frac {3}{2^22!} - 3\left(\frac {1}{2^33!} - \frac {1\cdot 3}{2^4 4!} + \frac {1\cdot 3 \cdot 5}{2^5 5!} + \cdots \right) \\ \implies S & = \frac 13 \left(1 + \frac 32 + \frac 38 - 2 \sqrt 2\right) \approx \boxed{0.01552} \end{aligned}$