How can I prove $$f(x)=\sum_{n=1}^\infty\dfrac{(-1)^n}{n}\chi_{[n-1,n)}(x)$$ is not Lebesgue Integrable?


  • $\begingroup$ math.stackexchange.com/questions/1095711/… $\endgroup$ – Yanko May 9 '20 at 15:28
  • 3
    $\begingroup$ You’re basically done, now you can actually do the integral on the right hand side to find $\sum \frac 1 n$, and the fact that your function isn’t integrable is equivalent to the divergence of the harmonic series. Of course, you should rigorously justify the swapping of the sum and integral; what theorem allows you to do that? $\endgroup$ – User8128 May 9 '20 at 15:46

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