Identity written as a sum of Möbius functions

Theorem

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.3.1 \mathrm{I}.B.$