prove-that-p-1-z-p-1-z-p-q-1-1-q-1-z-q-1-z-q-

Question Number 33358 by caravan msup abdo. last updated on 15/Apr/18

$${prove}\:{that}\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\frac{{z}^{{p}} }{\mathrm{1}+{z}^{{p}} }\:=\sum_{{q}=\mathrm{1}} ^{\infty} \:\left(−\mathrm{1}\right)^{{q}−\mathrm{1}} \frac{{z}^{{q}} }{\mathrm{1}−{z}^{{q}} } \\ $$

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