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Can u show that Lebesgue measurability of $$f(x)=\sum_{n=1}^\infty \frac{\cos nx}{n^2}$$ and $$\int\limits_0^\pi f(x)\,dx=\text{?}$$

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    $\begingroup$ One must suspect that the reason for the down-votes and the vote to close is that this question is phrased a bit too much in the style suitable for assigning homework. $\endgroup$ – Michael Hardy Feb 5 '15 at 3:16
  • $\begingroup$ For the first part of the question see here. $\endgroup$ – science Feb 5 '15 at 3:20
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The series converges to the pointwise limit of a sequence of continuous functions, so is measurable. The function $f$ is bounded by $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, so the dominated convergence theorem implies you can swap the sum and integral: $$\int_0^\pi f(x)\ \mathrm{d}x=\sum_{n=1}^\infty\int_0^\pi\frac{\cos nx}{n^2}\ \mathrm{d}x=\sum_{n=1}^\infty\frac1n\sin nx\bigg|_{x=0}^\pi=0.$$

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