Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing

This Question is Closed

UnkleRhaukusBest 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

UnkleRhaukusBest 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

UnkleRhaukusBest 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

UnkleRhaukusBest 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

NewOneBest 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

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

UnkleRhaukusBest 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

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

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

UnkleRhaukusBest ResponseYou've already chosen the best response.0
Thanks to both of you
 one year ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.