Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

If we put $$f(t) =\sum_{n=1}^{\infty}\frac{\sin nt} {n} $$ then $$f^{2}(t)=\sum_{n=1}^{\infty}\frac{\sin^{2}nt}{n^{2}} +\text{(terms containing }\sin nt\sin mt) $$ and integrating this term by term with respect to $t$ over $[-\pi, \pi] $ should give us $$\pi\sum_{n=1}^{\infty}\frac{1}{n^{2}}$$ and therefore we see that the RHS of the equation in question is $$\frac{3}{8\pi}\int_{-\pi}^{\pi}f^{2}(t)\,dt$$ This needs to be proved to be equal to $f^{2}(\pi/2)$. The function $f(t) $ is given in closed form as $$f(t) = \begin{cases} \dfrac{\pi - t} {2}, 0<t\leq \pi\\ 0,t=0\\ -\dfrac{\pi +t} {2}, - \pi\leq t<0 \end{cases}$$ and $f(t+2\pi)=f(t)$. So this works out fine.

Here is a proof of the identity $$\left(\sum_{n \geq 0} \frac{(-1)^n}{(2n+1)}\right)^2 = \frac{3}{8} \sum_{n \in \mathbb{N}} \frac{1}{n^2}$$ without knowing the value of either series.

Consider the double integral \begin{align*} I &= \int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x^2)(1+y^2)} \ dy \ dx. \end{align*} On one hand, we observe \begin{align*} \frac{1}{(1+x^2)(1+y^2)} &= \sum_{n \geq 0} \left(-x^2 \right)^n \sum_{m \geq 0} \left(-y^2 \right)^m, \quad |x|,|y| < 1. \end{align*} Then, we get that \begin{align*} I &= \int_{0}^1 \int_{0}^1 \sum_{n \geq 0} \left(-x^2 \right)^n \sum_{m \geq 0} \left(-y^2 \right)^m \ dy \ dx \\ &=\sum_{n \geq 0}\sum_{m \geq 0} \int_{0}^1 \int_{0}^1 \left(-x^2 \right)^n \left(-y^2 \right)^m \ dy \ dx \\ &=\sum_{n \geq 0}\sum_{m \geq 0} \frac{(-1)^n}{2n+1} \frac{(-1)^m}{2m+1} \\ &=\left(\sum_{n \geq 0} \frac{(-1)^n}{2n+1} \right)^2. \end{align*} The interchanging of sum and integral done here can be justified by the Dominated Convergence Theorem.

Now, we make the substitution $x \mapsto 1/x,$ and $y \mapsto 1/y$ to $I,$ resulting in \begin{align*} I &= \int_{1}^{\infty}\int_{1}^{\infty} \frac{1}{(1+x^2)(1+y^2)} \ dy \ dx, \end{align*} This implies \begin{align*} I &= \frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty} \frac{1}{(1+x^2)(1+y^2)} \ dy \ dx. \end{align*} Make another substitution $y=tx$ with $dy = x \ dt$ to see \begin{align*} I &= \frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty} \frac{x}{(1+x^2)(1+t^2x^2)} \ dt \ dx \\ &=\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty} \frac{x}{(1+x^2)(1+t^2x^2)} \ dx \ dt, \end{align*} in which the interchanging of the integral is justified by Tonelli's Theorem. Now, using the partial fractions identity, $$\frac{x}{(1+x^2)(1+t^2x^2)} = \frac{2 x}{\left(2-2 t^2\right) \left(x^2+1\right)}-\frac{2 t^2 x}{\left(2-2 t^2\right) \left(t^2 x^2+1\right)},$$ and integrating with respect to $x,$ we get \begin{align*} I &= \frac{1}{4}\int_{0}^{\infty} \frac{\log(t)}{t^2-1} \ dt. \end{align*} Now, observe \begin{align*} \int_{0}^{1} \frac{\log(t)}{t^2-1} \ dt + \int_{1}^{\infty} \frac{\log(t)}{t^2-1} \ dt= \int_{0}^{\infty} \frac{\log(t)}{t^2-1} \ dt. \end{align*} We see that \begin{align*} \int_{0}^{1} \frac{\log(t)}{t^2-1} \ dt &= \int_{1}^{\infty} \frac{\log(t)}{t^2-1} \ dt \end{align*} by making a change of variables $t \mapsto 1/t$ on the second integral. Thus, \begin{align*} I &= \frac{1}{2} \int_{0}^{1} \frac{\log(t)}{t^2-1} \ dt. \end{align*} Now, convert the integrand into a geometric series \begin{align*} \frac{\log(t)}{t^2-1} &= - \sum_{n \geq 0} \log(t) t^{2n}, \quad t \in (0,1), \end{align*} and we see that \begin{align*} I &= -\frac{1}{2} \int_{0}^{1} \sum_{n \geq 0} \log(t) t^{2n} \ dt \\ &= -\frac{1}{2} \sum_{n \geq 0} \int_{0}^{1} \log(t) t^{2n} \ dt \\ &= \frac{1}{2} \sum_{n \geq 0} \frac{1}{(2n+1)^2}, \end{align*} in which the interchanging of sum and integral is justified by Beppo Levi's Monotone Convergence Theorem. Observing that \begin{align*} \sum_{n \in \mathbb{N}} \frac{1}{n^2} &= \sum_{n \in \mathbb{N}} \frac{1}{(2n)^2}+\sum_{n \geq 0} \frac{1}{(2n+1)^2} \\ &= \frac{1}{4}\sum_{n \in \mathbb{N}} \frac{1}{n^2}+\sum_{n \geq 0} \frac{1}{(2n+1)^2}, \end{align*} we find \begin{align*} I &= \frac{3}{8} \sum_{n \in \mathbb{N}} \frac{1}{n^2}. \end{align*} Hence, we have $$\left(\sum_{n \geq 0} \frac{(-1)^n}{(2n+1)}\right)^2 = \frac{3}{8} \sum_{n \in \mathbb{N}} \frac{1}{n^2}.$$

Remark For those that may be interested, these types of integral calculations to evaluate series may be found in my joint paper with Daniele Ritelli. See https://www.ams.org/journals/qam/2018-76-03/S0033-569X-2018-01499-3/