Let's make the signs alternate

Algebra Level 2

What number goes into the blank below?

\[{\color{blue}1} + {\color{red}\dfrac12} + {\color{blue}\dfrac14} + {\color{red}\dfrac18} + {\color{blue}\dfrac1{16}} + \cdots =\left({\color{blue}1} - {\color{red}\dfrac12} + {\color{blue}\dfrac14} - {\color{red}\dfrac18} + {\color{blue}\dfrac1{16}} - \cdots \right) \times \text{_______} \]

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