Inspired by Sambhrant Sachan

Calculus Level 5

\[\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{4}{8}-\dfrac{9}{16}+\dfrac{11}{32}+\dfrac{56}{64}+\dfrac{1}{128}+\cdots+\dfrac{g(n)}{2^n}+\cdots\]

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.

Bonus: What if the series is

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

ceteris paribus?


Inspiration.

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