## anonymous one year ago Find the half range of cosine and sine expansion of: f(x) = { 1, 0<x<1/2 { 0, 1/2<x<1

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1. anonymous

The half range here is $$1$$, so the cosine and sine expansions are given by $2\int_0^1f(x)\cos nx\,dx\quad\text{and}\quad2\int_0^1f(x)\sin nx\,dx$(respectively). Compute these by splitting up the interval of integration; for example, $2\int_0^1f(x)\cos nx\,dx=2\left(\int_0^{1/2}\cos nx\,dx+\int_{1/2}^1 0\,dx\right)=2\int_0^{1/2}\cos nx\,dx$ Can you take it from here?