Legendre Polynomials & the Generating function

[dykuma]

Homework Statement

Homework Equations

and in chapter 1 I believe that wanted me to note that

The Attempt at a Solution

For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre polynomial will be odd. With odd Legendre polynomials, every coefficient has a term of x attributed to it (example P3(x) = 1/2(5x^3 - 3x)), so if it were to be evaluated at 0, the result will always be zero. However I am not sure how to prove that using the Generating function.

As for the second part of this question, I am not really sure what to do. For now I am looking for a place to start with that.

[EDIT] I did expand 5.1 as a Maclaurin Series, but I don't see how they want me to equate that to 5.2.

 

[dykuma]

I figured out the solution, just posting it here for sake of completeness.