study-the-convergence-of-k-0-e-i-kpi-x-and-find-its-sum-

Question Number 42187 by maxmathsup by imad last updated on 19/Aug/18

s t u d y t h e c o n v e r g e n c e o f \sum_{k = 0}^{\infty} e^{- i \frac{k π}{x}} a n d f i n d i t s s u m

Commented by maxmathsup by imad last updated on 22/Aug/18

l e t S = \sum_{k = 0}^{\infty} e^{- i \frac{k π}{x}} (w i t h x \neq 0)

S = \sum_{k = 0}^{\infty} {(e^{- i \frac{π}{x}})}^{k} = \frac{1}{1 - e^{- \frac{i π}{x}}} = \frac{1}{1 - c o s (\frac{π}{x}) + i s i n (\frac{π}{x})}

= \frac{1 - c o s (\frac{π}{x}) - i s i n (\frac{π}{x})}{{(1 - c o s (\frac{π}{x}))}^{2} + {s i n}^{2} (\frac{π}{x})} = \frac{1 - c o s (\frac{π}{x}) - i s i n (\frac{π}{x})}{2 - 2 c o s (\frac{π}{x})}

= \frac{1}{2} - \frac{i}{2} \frac{s i n (\frac{π}{x})}{2 {s i n}^{2} (\frac{π}{2 x})} = \frac{1}{2} - \frac{i}{2} \frac{2 s i n (\frac{π}{2 x}) c o s (\frac{π}{2 x})}{2 {s i n}^{2} (\frac{π}{2 x})} = \frac{1}{2} - \frac{i}{2} c o t a n (\frac{π}{2 x})

f i n a l l y S = \frac{1}{2} - \frac{i}{2} c o t a n (\frac{π}{2 x}) .