Integration of a function involving Legendre polynomials I am facing the following integration:
$\sum_{j=1}^{M} n \, \int_{0}^{\pi}P_{n-1}(\cos(\theta)) sin(j \,  \theta) d\theta$
With $n$ a positive integer, and with $M$ a positive ODD integer of course the summation of the $j$ index involves only ODD index ($1,3,5, \, ..., \, M$)
Could you help me in solving this integration? or perhaps giving me some hint? 
I thought that the terms:
$sin(\theta)+sin(3 \,  \theta)+sin(5 \,  \theta)+...+sin(M \,  \theta) $ 
could be expressed as a summation of (associated) Legendre polynomials but i'm struggling to show whether this is true or not. If this is the case perhaps the orthogonality of the Legendre polynomials might be used to work out the above integration.
Thanks in advance
 A: This is not an answer but it's too long for a comment.
Let $M=2N-1
 $. Then we can write the sum as $$\sum_{j\leq M,\, l\, odd}\sin\left(j\theta\right)=\sum_{k=0}^{N}\sin\left(\left(2k-1\right)\theta\right)
 $$ and the last sum has the closed form $$\sum_{k=0}^{N}\sin\left(\left(2k-1\right)\theta\right)=\frac{\sin^{2}\left(N\theta\right)}{\sin\left(\theta\right)}\tag{1}
  $$ and the RHS of $1$ can be expressed as sum of Legendre polynomials $$ \frac{\sin\left(N\theta\right)}{\sin\left(\theta\right)}=\sum_{l=0}^{N-1}P_{l}\left(\cos\left(\theta\right)\right)P_{N-1-l}\left(\cos\left(\theta\right)\right)
 $$ hence the integral is $$I=n\sum_{l=0}^{N-1}\int_{0}^{\pi}P_{n-1}\left(\cos\left(\theta\right)\right)P_{l}\left(\cos\left(\theta\right)\right)P_{N-1-l}\left(\cos\left(\theta\right)\right)\sin\left(N\theta\right)d\theta.
 $$ I don't know if it is useful, but maybe from this there is some way to go on. And maybe it's interesting to recall that holds $$\int_{0}^{\pi}P_{n-1}\left(\cos\left(\theta\right)\right)P_{l}\left(\cos\left(\theta\right)\right)P_{N-1-l}\left(\cos\left(\theta\right)\right)\sin\left(\theta\right)d\theta=2\left(\begin{array}{ccc}
n-1 & l & N-1-l\\
0 & 0 & 0
\end{array}\right)^{2}
 $$ where $\left(\begin{array}{ccc}
n-1 & l & N-1-l\\
0 & 0 & 0
\end{array}\right)
 $ is the $3j$ symbol. 
