find-the-sum-of-n-1-1-n-2-1-i-n-

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

f i n d t h e s u m o f \sum_{n = 1}^{\propto} \frac{1}{n} {(\frac{\sqrt{2}}{1 + i})}^{n} .

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

f o r z \in i R a n d ∣ z ∣⩽ 1 w e h a v e \sum_{n = 1}^{\propto} \frac{z^{n}}{n} = - l n (1 - z)

b u t w e h a v e \frac{\sqrt{2}}{1 + i} = \frac{1}{e^{i \frac{π}{4}}} = e^{- \frac{i π}{4}} w i t h ∣ e^{- \frac{i π}{4}} ∣= 1 s o

Σ \frac{1}{n} {(\frac{\sqrt{2}}{1 + i})}^{n} = - l n (1 - e^{- \frac{i π}{4}}) .

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

l n h e r e i s a c o m p l e x l o g a r i t h m e .