# Tweaking an easy problem

Algebra Level 5

$f(p,q)=\displaystyle\prod_{i=0}^{\infty} \left (1+\frac{1}{q^{p^i}}\right )$

If $$\displaystyle\sum_{n=2}^{\infty} \frac{\phi_1^n-\phi_2^n}{\phi_1^{2n}+\phi_2^{2n}-\phi_2^{n-1}\phi_1^{n-1}(\phi_2^2+\phi_1^2)}=\frac{a}{\sqrt{b}}$$ where $$b$$ is square free, and $$f(a,b)=\frac{\alpha}{\beta}$$, calculate $$\alpha+\beta$$.

Details and Assumptions

• $$\phi_1$$ is Golden Ratio which equals $$\frac{1+\sqrt{5}}{2}$$

• $$\phi_1\phi_2+1=0$$

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