Sum of totient equals z/((1-z) squared)

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Theorem

The following formula holds for $|z|<1$: $$\displaystyle\sum_{k=1}^{\infty} \dfrac{\phi(k)x^k}{1-x^k}= \dfrac{x}{(1-x)^2} ,$$ where $\phi$ denotes the totient.

Proof

References