How to find region of convergence of complex series $\sum\frac{(-1)^n}{z+n}$? According to me if z is any complex number then series is just like real alternating series so by Abel’s test of alternating series it is convergent . So according to me it’s is convergent in entire complex plane . Please suggest me exact reason . Thanks .
2026-04-02 16:57:05.1775149025
Region of convergence of complex series $\sum \frac{(-1)^n}{z+n}$.
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Couple together adjacent terms: $$-\frac1{z+2m-1}+\frac1{z+2m}=-\frac1{(z+2m-1)(z+2m)}.$$ We get a new series that converges absolutely, everywhere, save of course where the denominators vanish.