Find a recurrence relationhip for $a_{n}$:
$a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$
Where $f\left( x\right)= e^{-x}$
I have done it many times and keep obtianing zero. The hint in the question is to use Rodriguez' formula. And $P_{n}$ is a legendre polynomial.
Hint: Use Integration by parts, where you integrate $f(x)=e^{-x}$ and differentiate $P_n(x)$, then use the Rodrigues' formula.