# Sitting on the fence

Calculus Level 4

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

The numerators $$g(n)$$ are given by a linear recursion $$g(n)=ag(n-1)+bg(n-2)$$ for $$n>2$$, with $$g(1)=g(2)=1$$.

Find the value of the series above, to three significant figures. Enter 0.666 if you come to the conclusion that the series fails to converge.

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