Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$
Question:
Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$
Answer:
Let's think step by step.

user
\frac{11}{2}
Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$
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