Fourier Series/x over 0 to 2, x-2 over 2 to 4/Mistake

Source Work

 * : Chapter One: $\S 6$. Half-Range Cosine Series: Example $5$

Mistake

 * Find the half-range cosine series for
 * $\map f x = \begin{cases}

1, 0 < x < 2 \\ x - 2, 2 < x < 4 \end{cases}$
 * for the half-range $0 < x < 4$.

... and the required series is
 * $\map S x = \displaystyle 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \set {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } x \cos \frac {\paren {2 r - 1} \pi x} 4$.

Correction
The subsequent analysis is performed for the function:


 * $\map f x = \begin{cases}

x & : 0 < x < 2 \\ x - 2 & : 2 < x < 4 \end{cases}$

and the required series is actually:


 * $\map S x = \displaystyle 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \set {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } \cos \frac {\paren {2 r - 1} \pi x} 4$.