Integrate the product of sums!

Calculus Level 5

\[\displaystyle I = \int_0^\infty {\left(x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \right) \left(1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \right) \, dx}\]

The integral \(I \) above has a closed form. Find the value of this closed form.

Submit your answer as \(I^2\) to 2 decimal places.

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