The Problem: Consider the sign function on $(-1,0)\cup(0,1)$ defined by
$$ \sigma(x) := \left. \text{sgn}(x) \right|_{(-1,0)\cup(0,1)} = \begin{cases} 1 & x \in (0,1) \\ -1 & x \in (-1,0) \end{cases}$$
The problem is to show that
$$\int_{-1}^1 (\sigma(x))^2 dx = 2 \sum_{n=0}^\infty (4n+3) \left( \frac{(2n-1)!!}{(2n+2)!!} \right)^2$$
Context: This is (in its essence) problem $15.2.8$ in Mathematical Methods for Physicists by Arfken, Weber, & Harris. It was assigned to me as a homework problem for one of my classes. (In that vein I would prefer to only have nudges in the right direction, rather than full solutions.) The discussion in this section ($\S 15.2$) is essentially on Legendre polynomials and Fourier-Legendre series.
It is quite obvious that the integral evaluates to $2$, so the problem is ultimately showing that
$$\sum_{n=0}^\infty (4n+3) \left( \frac{(2n-1)!!}{(2n+2)!!} \right)^2 = 1$$
However, browsing the text, Wikipedia, and MathWorld don't give me any enlightening ideas on what identities to leverage. Expanding $f(x) = 1$ as a Fourier-Legendre series
$$f(x) = \sum_{n=0}^\infty c_n P_n(x) \; \text{where} \; c_n = \int_{-1}^1 f(x)P_n(x)dx$$
doesn't really lead me anywhere (for the integral in $c_n$ is zero whenever $n \ge 1$) - which is obvious enough, since $P_0(x) = 1$ anyways, so of course we'd get a finite series.
The identity does seem true. Taking the equivalent formulation of the problem (as a series equaling $1$) and summing $n=0$ to $n=100$ yields a result of about $0.996$ according to Wolfram, and up to $n=10,000$ yields about $0.999354$ (Wolfram), so it seems reasonable it converges to $1$, albeit somewhat slowly.
The original problem is in multiple parts: this is part (a), and part (c) notes, as I did, the integral $\int_{-1}^1 \sigma^2(x)dx = 2$. So it also seems plausible that I'm not even meant to calculate the integral at the outset, but instead utilize some other method. I suppose one could rewrite $\sigma$ as
$$ \sigma(x) = \begin{cases} P_0(x) & x \in (0,1) \\ -P_0(x) & x \in (-1,0) \end{cases}$$
and perhaps utilize some sort of identity used in the motivations/derivations tied to Legendre polynomials (a lot of integrals of $P_n^2$ seem to come up), but this rewriting doesn't give me anything more enlightening to work with.
Does anyone have some ideas as to how I might at least get started with this?