Fibonacci is Golden

Algebra Level 4

\[\large \lim_{n\to\infty}\sum_{k=1}^{2n} \left [ (-1)^k\frac{F_k}{\phi^k} \right ] \] Find the limit above, where \(F_k\) is the \(k\)-th Fibonacci number, \(\phi\) is the Golden Ratio, and \(n\) is a positive integer.

Answers in decimal form are accepted. If you find that the limit fails to exist, enter 0.666 as your answer.

Inspiration.

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