How to find the following summation

Question

How to find the following sum $$\sum\limits_{n=1}^{\infty}\dfrac{\left(\dfrac{3-\sqrt{5}}{2}\right)^n}{n^3}$$

I have tried by rationalizing. But after that I got stuck.

Answer 1

By definition, for $|r|<1$, $$ \sum_{n=1}^\infty r^n/n^3 = \text{polylog}(3,r)$$

This is not an elementary function.

Answer 2

For $x<1$,by differentiation,

$$S(x):=\sum_{n=1}^\infty\frac{x^n}{n^3}$$

$$xS'(x)=\sum_{n=1}^\infty\frac{x^n}{n^2}$$

$$x(xS'(x))'=\sum_{n=1}^\infty\frac{x^n}{n}$$

$$(x(xS'(x))')'=\sum_{n=1}^\infty x^{n-1}=\frac1{1-x}.$$

Then by integration

$$(xS'(x))'=-\frac{\log(1-x)}x$$ and the closed-form stops here.