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The title is pretty self-explanatory. What is the value of $\sum_{n=1}^{\infty}\frac{f_n}{n^2}$

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$f_n$ grows exponentially, so the sum diverges. To elaborate, if $a_n := \dfrac{f_n}{n^2}$, it can be shown that $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| = \varphi = \frac{1+\sqrt{5}}{2} > 1,$$ and so by the ratio test, the series diverges.

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