Sum : $\\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n+1)^3}$

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Then we can express $f$ in a Fourer series: \begin{equation*} f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n \cos n\pi t+b_n \sin n\pi t \end{equation*} where \begin{align*} &a_n=\frac{1}{1}\int_{0}^{2}f(t)\cos n\pi t \, dt \\ &b_n=\frac{1}{1}\int_{0}^{2}f(t)\sin n\pi t\, dt \end{align*} But $f$ is odd, so $a_n=0$.

Related Sum : $\\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n+1)^3}$