let-S-p-n-0-cos-npi-p-and-W-p-n-0-sin-npi-p-with-p-natural-integr-not0-1-find-a-simple-form-of-S-p-and-W-p-2-find-the-value-of-n-0-cos-npi-3-and-n-0-

Question Number 42285 by maxmathsup by imad last updated on 22/Aug/18

l e t S_{p} = \sum_{n = 0}^{\infty} c o s (\frac{n π}{p}) a n d W_{p} = \sum_{n = 0}^{\infty} s i n (\frac{n π}{p}) w i t h p n a t u r a l i n t e g r n o t 0

1) f i n d a s i m p l e f o r m o f S_{p} a n d W_{p}

2) f i n d t h e v a l u e o f \sum_{n = 0}^{\infty} c o s (\frac{n π}{3}) a n d \sum_{n = 0}^{\infty} s i n (\frac{n π}{3})

3) f i n d t h e v a l u e o f \sum_{n = 0}^{\infty} c o s (\frac{n π}{5}) a n d \sum_{n = 0}^{\infty} s i n (\frac{n π}{5})

4) c a l c u l a t e A = \sum_{n = 0}^{\infty} {c o s}^{2} (\frac{n π}{3}) a n d B = \sum_{n = 0}^{\infty} {s i n}^{2} (\frac{n π}{3}) .

Commented by maxmathsup by imad last updated on 23/Aug/18

1) w e h a v e S_{p} + {i W}_{p} = \sum_{n = 0}^{\infty} e^{i \frac{n π}{p}} = \sum_{n = 0}^{\infty} {(e^{\frac{i π}{p}})}^{n} = \frac{1}{1 - e^{i \frac{π}{p}}}

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

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

= \frac{i}{2 s i n (\frac{π}{2 p})} {c o s (\frac{π}{2 p}) - i s i n (\frac{π}{2 p})} = \frac{i}{2} c o t a n (\frac{π}{2 p}) + \frac{1}{2} \Rightarrow

S_{p} = \frac{1}{2} a n d W_{p} = \frac{1}{2} c o t a n (\frac{π}{2 p})

2) \sum_{n = 0}^{\infty} c o s (\frac{n π}{3}) = S_{3} = \frac{1}{2} a n d \sum_{n = 0}^{\infty} s i n (\frac{n π}{3}) = W_{3} = \frac{1}{2} c o t a n (\frac{π}{6})

W_{3} = \frac{1}{2} \sqrt{3} = \frac{\sqrt{3}}{2}

3) \sum_{n = 0}^{\infty} c o s (\frac{n π}{5}) = S_{5} = \frac{1}{2} a n d \sum_{n = 0}^{\infty} s i n (\frac{n π}{5}) = W_{5} = \frac{1}{2} c o t a n (\frac{π}{10})

Commented by maxmathsup by imad last updated on 23/Aug/18

4) t h e Q i s c l a c u l a t e A_{n} = \sum_{k = 0}^{n} {c o s}^{2} (\frac{k π}{3}) a n d B_{n} = \sum_{k = 0}^{n} {s i n}^{2} (\frac{k π}{3}) .