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Question Number 27912 by abdo imad last updated on 16/Jan/18

$$findthesumof\sum _{n=1}^{\propto }\frac{1}{n}{\left(\frac{\sqrt{2}}{1+i}\right)}^n.$$

Commented by abdo imad last updated on 23/Jan/18

$$forz\in iRand\mid z\mid ⩽1wehave\sum _{n=1}^{\propto }\frac{z^n}{n}=-ln(1-z)$$$$butwehave\frac{\sqrt{2}}{1+i}=\frac{1}{e^{i\frac{\pi }{4}}}=e^{-\frac{i\pi }{4}}with\mid e^{-\frac{i\pi }{4}}\mid =1so$$$$\mathrm{\Sigma }\frac{1}{n}{\left(\frac{\sqrt{2}}{1+i}\right)}^n=-ln(1-e^{-\frac{i\pi }{4}}).$$

Commented by abdo imad last updated on 23/Jan/18

$$lnhereisacomplexlogarithme.$$

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