Identity written as a sum of Möbius functions

Theorem

The following formula holds for $|x|<1$: $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)x^k}{1-x^k} = x,$$ where $\mu$ denotes the Möbius function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 24.3.1 I.B.