<s>[INST] Evaluate $\sum\limits_{n=1}^{+\infty} \frac{\left( \frac{3-\sqrt{5}}{2} \right)^{n}}{n^{3}}$ Evaluate
$$\sum\limits_{n=1}^{+ \infty} \frac{  \left( \frac{3-\sqrt{5}}{2} \right)^{n} }{n^{3}}$$
We can use the Fourier series to calculate this sum, because it converges.
Also, we know that $\frac{3-\sqrt{5}}{2} = \frac{1}{\varphi^{2}}$ where $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. What is going on about this number $\frac{3-\sqrt{5}}{2}$ ? we know something else ?

Thank you for your answer but we know something else without about trilogarithm function?
 [/INST] $$\sum_{n=1}^\infty \frac{a^n}{n^3}=\text{Li}_3(a)$$ Making $a=\frac{3-\sqrt{5}}{2} = \frac{1}{\varphi^{2}}$, you just get a number
$$\text{Li}_3\left(\frac{1}{\varphi ^2}\right)=0.4026839629521090211599594481825111422197338\cdots$$
</s>