Infinite sum/difference of ratios of products

\[ 1 - \frac{ 3 } { 4 } + \frac{3 \times 5 } { 4 \times 8 } - \frac{ 3 \times 5 \times 7 } { 4 \times 8 \times 12 } + \frac{ 3 \times 5 \times 7 \times 9 } { 4 \times 8 \times 12 \times 16 } - \cdots \, ?\]

What is the exact value of the expression above?

Give your answer to 3 decimal places.

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