1) simplify S_n (x)=Σ_(k=1) ^n sin^2 (kx) 2)simplify A_n =Σ


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 39332 by abdo mathsup 649 cc last updated on 05/Jul/18

$1) s i m p l i f y S_{n} (x) = \sum_{k = 1}^{n} {s i n}^{2} (k x)$ $2) s i m p l i f y A_{n} = \sum_{k = 1}^{n} {s i n}^{2} (\frac{k π}{n})$
Commented by math khazana by abdo last updated on 09/Jul/18
$1) we have S_n (x)=Σ_(k=0) ^n ((1−cos(2kx))/2) = ((n+1)/2) −(1/2) Σ_(k=0) ^n cos(2kx) but Σ_(k=0) ^n cos(2kx) = Re(Σ_(k=0) ^n e^(i2kx) ) Σ_(k=0) ^n e^(i2kx) =Σ_(k=0) ^n (e^(i2x) )^k = ((1− e^(i2(n+1)x) )/(1−e^(i2x) )) =((1 −cos2(n+1)x −i sin2(n+1)x)/(1−cos(2x)−i sin(2x))) = ((2sin^2 (n+1)x −2i sin(n+1)x cos(n+1)x)/(2sin^2 x −2i sinx cosx)) =((−isin(n+1)x{cos(n+1)x +isin(n+1)x})/(−isinx{cosx +isinx})) =((sin(n+1)x)/(sinx)) e^(i(n+1)x) e^(−ix) = ((sin(n+1)x)/(sinx)) {cos(nx) +i sinx} ⇒ Σ_(k=0) ^n cos(2kx) = ((sin(n+1)x)/(sinx)) cos(nx) ⇒ S_n (x) =((n+1)/2) −((sin(n+1)x cos(nx))/(2sinx)) . 2) A_n =S_n ((π/n)) =((n+1)/2) −((sin(n+1)(π/n) cos(n(π/n)))/(2sin((π/n)))) =((n+1)/2) + ((sin(π +(π/n)))/(2sin((π/n)))) =((n+1)/2) +((−sin((π/n)))/(2sin((π/n)))) =((n+1)/2) −(1/2) = (n/2) ⇒ A_n = (n/2) .$
$1) w e h a v e S_{n} (x) = \sum_{k = 0}^{n} \frac{1 - c o s (2 k x)}{2}$ $= \frac{n + 1}{2} - \frac{1}{2} \sum_{k = 0}^{n} c o s (2 k x) b u t$ $\sum_{k = 0}^{n} c o s (2 k x) = R e (\sum_{k = 0}^{n} e^{i 2 k x})$ $\sum_{k = 0}^{n} e^{i 2 k x} = \sum_{k = 0}^{n} {(e^{i 2 x})}^{k} = \frac{1 - e^{i 2 (n + 1) x}}{1 - e^{i 2 x}}$ $= \frac{1 - c o s 2 (n + 1) x - i s i n 2 (n + 1) x}{1 - c o s (2 x) - i s i n (2 x)}$ $= \frac{2 {s i n}^{2} (n + 1) x - 2 i s i n (n + 1) x c o s (n + 1) x}{2 {s i n}^{2} x - 2 i s i n x c o s x}$ $= \frac{- i s i n (n + 1) x {c o s (n + 1) x + i s i n (n + 1) x}}{- i s i n x {c o s x + i s i n x}}$ $= \frac{s i n (n + 1) x}{s i n x} e^{i (n + 1) x} e^{- i x}$ $= \frac{s i n (n + 1) x}{s i n x} {c o s (n x) + i s i n x} \Rightarrow$ $\sum_{k = 0}^{n} c o s (2 k x) = \frac{s i n (n + 1) x}{s i n x} c o s (n x) \Rightarrow$ $S_{n} (x) = \frac{n + 1}{2} - \frac{s i n (n + 1) x c o s (n x)}{2 s i n x} .$ $2) A_{n} = S_{n} (\frac{π}{n}) = \frac{n + 1}{2} - \frac{s i n (n + 1) \frac{π}{n} c o s (n \frac{π}{n})}{2 s i n (\frac{π}{n})}$ $= \frac{n + 1}{2} + \frac{s i n (π + \frac{π}{n})}{2 s i n (\frac{π}{n})} = \frac{n + 1}{2} + \frac{- s i n (\frac{π}{n})}{2 s i n (\frac{π}{n})}$ $= \frac{n + 1}{2} - \frac{1}{2}$ $= \frac{n}{2} \Rightarrow A_{n} = \frac{n}{2} .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum