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UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[\qquad\text{\(f(x)\) as a sine series}\] \begin{equation*} f(x)=\cos(x),\qquad0<x<\pi \end{equation*} \begin{equation*} % g(x) g(x)= \begin{cases} \cos(x),&0<x<\pi\\ \cos(x),&\pi<x<0 \end{cases}\qquad\text{odd extension} \end{equation*}
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[\begin{align*} % a_0,n a_0&=a_n=0\qquad\text{(odd function)}\\ \end{align*}\] \begin{align*} % b_n b_n&=\frac1\pi\int\limits_{\pi}^\pi g(x)\sin(nx)\,\text dx\\ &=\frac2\pi\int\limits_{0}^\pi \cos(x)\sin(nx)\,\text dx\qquad\text{(even integrand)}\\ &=\frac1\pi\int\limits_{0}^\pi \sin\big((1+n)x\big)\sin\big((1n)x\big)\,\text dx\\ &=\frac1\pi\left[\frac{\cos\big((1+n)x\big)}{1+n}+\frac{\cos\big((1n)x\big)}{1n}\Big_0^\pi\right]\\ &=\frac1\pi\left[\frac{1\cos\big((1+n)\pi\big)}{1+n}+\frac{\cos\big((1n)\pi\big)1}{1n}\right]\\ &=\frac1\pi\left[\frac{1(1)^{n+1}}{1+n}+\frac{(1)^{n+1}1}{1n}\right]\\ &=\frac1\pi\left[\frac{1+(1)^{n}}{1+n}+\frac{(1)^{n+1}1}{1n}\right]\\ &=\frac1\pi\left[\frac{(1n)\big(1+(1)^n\big)(1+n)\big((1)^{n}+1\big)}{1n^2}\right]\\ &=\frac1\pi\left[\frac{2n\big((1)^{n}+1\big)}{1n^2}\right]\\ &=\frac2\pi\left[\frac{n\big((1)^{n}+1\big)}{n^21}\right]\\ \end{align*}
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[\begin{align*} % S(x) S(x)&=\frac2\pi\sum\limits_{n=1}^\infty\frac{n\big((1)^{n}+1\big)}{n^21}\sin(nx)\\ &=\frac4\pi\sum\limits_{n=2,4,6,\dots}^\infty\frac{n\sin(nx)}{n^21}\\ &=\frac8\pi\sum\limits_{r=1}^\infty\frac{r\sin(2rx)}{4r^21}\\ \end{align*}\]
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
for some reason in the back of my book it has \[\frac8\pi\sum\limits_{r=1}^\infty\frac{r\sin(2rx)}{4r^2+1}\] as the answer, which seams like a mistake to me , but maybe i have overlooked something
 one year ago

NewOne Group TitleBest ResponseYou've already chosen the best response.1
The math checks out.
 one year ago

NewOne Group TitleBest ResponseYou've already chosen the best response.1
He used the last trig identity on this page convert it. http://www.sosmath.com/trig/Trig5/trig5/trig5.html
 one year ago

NewOne Group TitleBest ResponseYou've already chosen the best response.1
Were you expecting bn as 0?
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
the only qualm i have on this question is the plus sign in the denominator of answer in the back of the book, i got a minus sign , which seams to graph the correct thing so im not sure .
 one year ago

NewOne Group TitleBest ResponseYou've already chosen the best response.1
I got minus as well.
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
i guess the back of my book is wrong then ,
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.0
yeah, even i get minus doing it another way also...
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
Thanks to both of you
 one year ago
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