Tetration And Telescoping

Calculus Level 5

\[ \large \sum_{x=2}^\infty \sum_{n=0}^\infty \dfrac{2^n}{1 + x^{2^n}} (-1)^x \]

The iterated series above has a closed form.

Let this closed form be denoted by \(\omega\), submit your answer as \( \lfloor 1000 \omega \rfloor \).

\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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