8: For the function of length L = 2, the sine and cosine expansions have double the length of period, that is T = 2L = 4, and the special frequency ω = π/2. Probably the easiest way to find the necessary coefficients is to use the standard formula, with L in place of T and the new special ω.

The sine series has a_k = 0. For the other coefficients you should get (using integration by parts)

Use them to create the appropriate series. Then draw the odd periodic extension of f and apply Jordan's conditions to find the sum of this series.

The cosine series has b_k = 0. For the other coefficients you should get (using integration by parts)

Use them to create the appropriate series. Then draw the even periodic extension of f and apply Jordan's conditions to find the sum of this series.

Answer