A function f(t) is defined for 0 ≤ t ≤ 1/2T by
$$f\left(t\right)=\{\begin{array}{cc}t& (0⩽t⩽\frac{1}{4}T)\\ \frac{1}{2}T-t& (\frac{1}{4}T⩽t⩽\frac{1}{2}T)\end{array}$$Sketch odd and even functions that have a period T and are equal to f (t) for 0 ≤ t ≤ 1/2T.

(a) Find the half-range Fourier sine series of f(t).

(b) To what value will the series converge for t = − 1/4T?

(c) What is the sum of the following series?

1 s = (2r - 1) r=1