In each of the following sums, shift the index $n\rightarrow n+2$. Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:
$$\sum_{n=0}^3 n$$
$$\sum_{n=1}^5 e^{in\phi}$$
$$\sum_{n=0}^{\infty} a_n n(n-1)z^{n-2}$$
By hand, find the recurrence relation for a power series solution $H(\rho)$ of the equation:
$$\rho {d^2 H\over d\rho^2} +(2\ell+2-\rho){dH\over d\rho} +(\lambda-\ell-1) H=0$$
where $\ell$ is a known positive integer, and $\lambda$ is an unknown constant.
Suppose that you want a solution to (a) which is a polynomial of degree 4. Assume that $\ell=2$. What does that tell you about the unknown constant $\lambda$?
Find the polynomial of degree 4 solution to the differential equation in part (a) assuming $\ell=2$. Assume anything you need to about $\lambda$.
Use your favorite tool (\emph{e.g.} Maple, Mathematica, Matlab, pencil) to generate the Legendre polynomial expansion to the function $f(z)=\sin(\pi z)$. How many terms do you need to include in a partial sum to get a “good” approximation to $f(z)$ for $-1<z<1$? What do you mean by a “good” approximation? How about the interval $-2<z<2$? How good is your approximation? Discuss your answers. Answer the same set of questions for the function $g(z)=\sin(3\pi z)$