# Inspired by Fibonacci et al

Calculus Level 5

$\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{3}{8}-\dfrac{7}{16}+\dfrac{5}{32}+\dfrac{33}{64}+\dfrac{13}{128}+\cdots+\dfrac{g(n)}{2^n}+\cdots$

The numerators $$g(n)$$ are given by $$g(n)=g(n-1)-4g(n-2)$$ for $$n>2$$, with $$g(1)=g(2)=1$$.

If $$M$$ and $$m$$ are the supremum and the infimum, respectively, of the partial sums of the series above, find $$M+m$$. As your answer, enter $$\lfloor 1000(M+m) \rfloor$$.

Enter 2016 if you come to the conclusion that the supremum or the infimum fail to exist.



Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

This problem is a follow-up to this question.

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