1)simplify S_n (x)=Σ_(k=0) ^n C_n ^k cos^k (x)cos(kx) 2)find the value of A_n =Σ_(k=0) ^n C


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Question Number 66462 by mathmax by abdo last updated on 15/Aug/19

$1) s i m p l i f y S_{n} (x) = \sum_{k = 0}^{n} C_{n}^{k} {c o s}^{k} (x) c o s (k x)$ $2) f i n d t h e v a l u e o f A_{n} = \sum_{k = 0}^{n} C_{n}^{k} {c o s}^{k} (\frac{π}{n}) c o s (\frac{k π}{n})$
Commented by mathmax by abdo last updated on 21/Aug/19
$1) we have S_n (x)=Re( Σ_(k=0) ^n C_n ^k cos^k x e^(ikx) ) but Σ_(k=0) ^n C_n ^k cos^k xe^(ikx) =Σ_(k=0) ^n C_n ^k (e^(ix) cosx)^k ×1^(n−k) =(1+e^(ix) cosx)^n =(1+cosx{cosx +isinx})^n =(1+cos^2 x +isinx cosx)^n let z =1+cos^2 x +isinx cosx ⇒∣z∣=(√((1+cos^2 x)^2 +sin^2 x cos^2 x)) =(√(1+cos^4 x +2cos^2 x +(1−cos^2 x)cos^2 x)) =(√(1+cos^4 x +2cos^2 x+cos^2 x−cos^4 x))=⇒(√(1+3cos^2 x)) ⇒ z =(√(1+3cos^2 x))e^(i arctan(((sinx cosx)/(1+cos^2 x)))) we have ((sinxcosx)/(1+cos^2 x)) =((sin(2x))/(2(1+((1+cos(2x))/2)))) =((sin(2x))/(2+1+cos(2x))) =((sin(2x))/(3+cos(2x))) and 1+cos^2 x =1+((1+cos(2x))/2) =((3+cos(2x))/2) ⇒ z =(((3+cos(2x))/2))^(1/2) e^(i arctan(((sin(2x))/(3+cos(2x))))) ⇒ z^n =(((3+cos(2x))/2))^(n/2) e^(in arctan(((sin(2x))/(3+cos(2x))))) ⇒ S_n (x) =Re(z^n ) =(((3+cos(2x))/2))^(n/2) cos(n arctan(((sin(2x))/(3+cos(2x))))) 2) we have A_n =Σ_(k=0) ^n C_n ^k cos^k ((π/n)) cos(((kπ)/n)) =S_n ((π/n)) =(((3+cos(((2π)/n)))/2))^(n/2) cos(narctan(((sin(((2π)/n)))/(3+cos(((2π)/n)))))).$
$1) w e h a v e S_{n} (x) = R e (\sum_{k = 0}^{n} C_{n}^{k} {c o s}^{k} x e^{i k x}) b u t$ $\sum_{k = 0}^{n} C_{n}^{k} {c o s}^{k} {x e}^{i k x} = \sum_{k = 0}^{n} C_{n}^{k} {(e^{i x} c o s x)}^{k} \times 1^{n - k} = {(1 + e^{i x} c o s x)}^{n}$ $= {(1 + c o s x {c o s x + i s i n x})}^{n} = {(1 + {c o s}^{2} x + i s i n x c o s x)}^{n}$ $l e t z = 1 + {c o s}^{2} x + i s i n x c o s x \Rightarrow∣ z ∣= \sqrt{{(1 + {c o s}^{2} x)}^{2} + {s i n}^{2} x {c o s}^{2} x}$ $= \sqrt{1 + {c o s}^{4} x + 2 {c o s}^{2} x + (1 - {c o s}^{2} x) {c o s}^{2} x}$ $= \sqrt{1 + {c o s}^{4} x + 2 {c o s}^{2} x + {c o s}^{2} x - {c o s}^{4} x} =\Rightarrow \sqrt{1 + 3 {c o s}^{2} x} \Rightarrow$ $z = \sqrt{1 + 3 {c o s}^{2} x} e^{i a r c t a n (\frac{s i n x c o s x}{1 + {c o s}^{2} x})}$ $w e h a v e \frac{s i n x c o s x}{1 + {c o s}^{2} x} = \frac{s i n (2 x)}{2 (1 + \frac{1 + c o s (2 x)}{2})} = \frac{s i n (2 x)}{2 + 1 + c o s (2 x)} = \frac{s i n (2 x)}{3 + c o s (2 x)}$ $a n d 1 + {c o s}^{2} x = 1 + \frac{1 + c o s (2 x)}{2} = \frac{3 + c o s (2 x)}{2} \Rightarrow$ $z = {(\frac{3 + c o s (2 x)}{2})}^{\frac{1}{2}} e^{i a r c t a n (\frac{s i n (2 x)}{3 + c o s (2 x)})} \Rightarrow$ $z^{n} = {(\frac{3 + c o s (2 x)}{2})}^{\frac{n}{2}} e^{i n a r c t a n (\frac{s i n (2 x)}{3 + c o s (2 x)})} \Rightarrow$ $S_{n} (x) = R e (z^{n}) = {(\frac{3 + c o s (2 x)}{2})}^{\frac{n}{2}} c o s (n a r c t a n (\frac{s i n (2 x)}{3 + c o s (2 x)}))$ $2) w e h a v e A_{n} = \sum_{k = 0}^{n} C_{n}^{k} {c o s}^{k} (\frac{π}{n}) c o s (\frac{k π}{n})$ $= S_{n} (\frac{π}{n}) = {(\frac{3 + c o s (\frac{2 π}{n})}{2})}^{\frac{n}{2}} c o s (n a r c t a n (\frac{s i n (\frac{2 π}{n})}{3 + c o s (\frac{2 π}{n})})) .$
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