I've got a problem that I have been working on in my Real Analysis class, and am not sure on the answer. The problem is below along with my thoughts so far.
Problem:
Let f be a function such that $f(x) = \frac{(-1)^n}{n}$ for x $\in [n, n+1)$. Is f Lebesgue integrable on $[1,\infty)$?
My thoughts:
In class we learned that a function is Lebesgue integrable if $\int_E f^+$ and $\int_E f^-$ are finite, where $f^+=max\{f,0\}$ and $f^-=max\{-f,0\}$.
Since it can be shown that $$\int_{[1,\infty)}f^+=\lim_{n \to \infty} \int_{[1,n]}f^+=lim_{n \to \infty} \sum_{k=1}^n\frac{1}{2k}$$ and the sum diverges, I am thinking that f is not Lebesgue integrable. Is my reasoning on this correct?
Indeed, $f$ is Lebesgue integrable if and only if $|f|$ is, because as you mention, you define the integral using $f^+$ and $f^-$, both of which have to be finite.
Since $|f|=f^++f^-$, the function you give has $ |f(x)|= \frac{1}{n}$ on $[n,n+1)$, and we know $\sum_{n} \frac{1}{n}$ diverges.