find-k-0-n-C-n-k-cos-kpi-n-

Question Number 90749 by abdomathmax last updated on 25/Apr/20

f i n d \sum_{k = 0}^{n} C_{n}^{k} c o s (\frac{k π}{n})

Commented by mathmax by abdo last updated on 26/Apr/20

A_{n} = \sum_{k = 0}^{n} C_{n}^{k} c o s (\frac{k π}{n}) \Rightarrow A_{n} = R e (\sum_{k = 0}^{n} C_{n}^{k} e^{\frac{i k π}{n}})

w e h a v e \sum_{k = 0}^{n} C_{n}^{k} e^{\frac{i k π}{n}} = \sum_{k = 0}^{n} C_{n}^{k} {(e^{\frac{i π}{n}})}^{k}

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

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

= {(2 c o s (\frac{π}{2 n}) e^{\frac{i π}{2 n}})}^{n} = 2^{n} {c o s}^{n} (\frac{π}{2 n}) e^{i \frac{π}{2}} = 2^{n} {c o s}^{n} (\frac{π}{2 n}) i \Rightarrow A_{n} = 0