In Problems 9.2, of Chapter 9, of Differential Equations and Boundary Value Problems (Fifth Edition) by C. Henry Edwards and David Penney and David T. Calvis, the following is the Question # 10:
Find the Fourier series of the following function (given in one full period) $$ f(t) = \begin{cases} 0 & \text{if } -2 \lt t \lt 0 \\ t^2 & \text{if } \quad 0 \lt t \lt 2 \end{cases} $$
My solution for the Fourier coefficients are as follows: $$ a_n = \frac{1}{2} \int_{-2}^{2} f(t) \cos \frac{n \pi}{2} t ~ dt\\ a_n = \frac{1}{2} \int_{0}^{2} t^2 \cos \frac{n \pi}{2} t ~dt \\ \textrm{applying integrations by parts repeatedly}\\ a_n = \frac{8}{n^2 \pi^2} (-1)^n $$
Similarly, $$ b_n = \frac{1}{2} \int_{0}^{2} t^2 \sin \frac{n \pi}{2} t ~dt \\ \textrm{repeatedly applying integration by parts as before}\\ b_n = \frac{1}{2} \left( \frac{-2}{n \pi} \times 4 (-1)^n + \frac{16}{n^3 \pi^3} (-1)^n \right)\\ b_n = \frac{-4}{n \pi} (-1)^n + \frac{16}{n^3 \pi^3} (-1)^n $$
$$ a_0 = \frac{1}{2}\int_{0}^{2} t^2 dt = \frac{1}{6} t^3 \big|_{0}^{3}= \frac{4}{3} $$
$$ a_n = \begin{cases} \dfrac43 & \text{if } n=0 \\\\ \dfrac{8}{n^2 \pi^2} (-1)^n & \text{if } n \geq 1 \end{cases} $$
$$ b_n = \frac{-4}{n \pi} (-1)^n + \frac{16}{n^3 \pi^3} (-1)^n $$
and
$$ \begin{align} \textrm{ Fourier Series} = \color{purple}{\frac{2}{3}} &+ \color{purple}{\frac{-8}{\pi^2} \left( \cos \frac{\pi}{2} t - \frac{1}{4} \cos \frac{2\pi}{2} t + \frac{1}{9} \cos \frac{3 \pi}{2} t - \cdots \right)} + \\ &+ \color{green}{ \frac{4}{\pi} \left( \sin \frac{\pi}{2} t - \frac{1}{2} \sin \frac{2 \pi}{2} t + \frac{1}{3} \sin \frac{3 \pi}{2} t - \cdots \right) } -\\ &- \color{red}{ \frac{16}{\pi^3} \left( \sin \frac{\pi}{2} t - \frac{1}{8} \sin \frac{2 \pi}{2} t - \frac{1}{27} \sin \frac{3\pi}{2} t + \cdots \right)} \end{align} $$
However, the answer given in the book is as follows. The mismatch are the terms in the bracket following the coefficient $\frac{16}{\pi^3}$, the books lists, I don't know why, only the odd $n$ s. Where is my mistake?
