(a) Consider the following function and its periodic extension: f(x)=sinx, where 2

(a) Consider the following fimction and its periodic extension: f(x) Isin , where and F (x) where nez Find the Fourier seriee; representation of Note that L _ (b) Does the FVnrier serifs you found in (a) converge, uniformly? If so, prove this using the Weierstrass test for uniform If not, discuss whether it converges pointwise anywhere, in its (c) Use your result in (a) to evaluate the following two series (d) Attempt to differentiate your Fourier series in (a) term by term. Does the series uniformly? If so, prove this using the test for uniform convergence. If not, discrss whether it pointwise anywhere in its domain. (e) Carefully differentiate the function F(x), as defined in (a), and then find the Fourier series of the resulting fimction, i.e., find the Fourier serif; of V (f) Write 1-2 sentences comparing your series' from (d) and discnwing how well the term by term differentiation hms worked. (g) Attempt to differentiate your Fourier series in (d) term by term. Dot* the, resulting serif; uniformly? If so, prove this using the Weierst.r;s test for uniform If not, discuss whether it converges pointwise anywhere in its domain. (h) Carefully differentiate your answer from (e), and then find the Fourier series of the resulting function, i.e., find the Fourier serif; of You may reuse working from previous questions. (i) Write 1-2 sentences comparing your series' from (g) and (h), discussing how well the term by term differentiation hiv; worked.