Question

cos(x) as a sine series

Answer 1

\[\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*}

Answer 2

\[\begin{align*} % a_0,n
a_0&=a_n=0\qquad\text{(odd function)}\\
\end{align*}\]

\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((1-n)x\big)\,\text dx\\
&=\frac1\pi\left[\frac{-\cos\big((1+n)x\big)}{1+n}+\frac{\cos\big((1-n)x\big)}{1-n}\Big|_0^\pi\right]\\
&=\frac1\pi\left[\frac{1-\cos\big((1+n)\pi\big)}{1+n}+\frac{\cos\big((1-n)\pi\big)-1}{1-n}\right]\\
&=\frac1\pi\left[\frac{1-(-1)^{n+1}}{1+n}+\frac{(-1)^{n+1}-1}{1-n}\right]\\
&=\frac1\pi\left[\frac{1+(-1)^{n}}{1+n}+\frac{(-1)^{n+1}-1}{1-n}\right]\\
&=\frac1\pi\left[\frac{(1-n)\big(1+(-1)^n\big)-(1+n)\big((-1)^{n}+1\big)}{1-n^2}\right]\\
&=\frac1\pi\left[\frac{-2n\big((-1)^{n}+1\big)}{1-n^2}\right]\\
&=\frac2\pi\left[\frac{n\big((-1)^{n}+1\big)}{n^2-1}\right]\\
\end{align*}

Answer 3

\[\begin{align*} % S(x)
S(x)&=\frac2\pi\sum\limits_{n=1}^\infty\frac{n\big((-1)^{n}+1\big)}{n^2-1}\sin(nx)\\
&=\frac4\pi\sum\limits_{n=2,4,6,\dots}^\infty\frac{n\sin(nx)}{n^2-1}\\
&=\frac8\pi\sum\limits_{r=1}^\infty\frac{r\sin(2rx)}{4r^2-1}\\
\end{align*}\]

Answer 4

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

Answer 5

The math checks out.

Answer 6

He used the last trig identity on this page convert it. http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Answer 7

Were you expecting bn as 0?

Answer 8

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 .

Answer 9

I got minus as well.

Answer 10

i guess the back of my book is wrong then ,

Answer 11

yeah, even i get minus doing it another way also...

Answer 12

Thanks to both of you