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I am trying to understand the following example from Apostol's Mathematical Analysis (p.275).

Let $f(x):[0,\infty)\rightarrow \mathbb{R}$ be defined by $$ f(x)=\frac{(-1)^n}{n}, \,\,\,\, x\in [n-1,n), \,\,\,\, n=1,2,3,\ldots $$ From the graph of the function, we see that the area estimation is essentially $-1+\frac{1}{2}-\frac{1}{3}+\cdots$ which is $-\log 2$.

But, I could not realize why the function is not integrable on $[0,\infty)$ (as Apostol asserts).

If we pick a positive point $b$ on real line, there will be some integer $n$ just behind it, so area bounded by $f$ from $0$ to $b$ will be between $-1+\frac{1}{2}-\cdots +\frac{(-1)^n}{n}$ and $-1+\frac{1}{2}-\cdots +\frac{(-1)^{n+1}}{n+1}$ and these two series are converging to $-\log 2$. So how the author is asserting that $f$ is not integrable on $[0,\infty)$?

Anne Bauval
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Maths Rahul
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  • Integrability takes absolute value into consideration. – mowzorn Dec 29 '22 at 07:02
  • Butt $|f|$ is integrable implies $f$ is integrable, right? (or am I missing something? Similar to convergence of series, when we conseider convergence of series, absolute convergence implies convergence.) – Maths Rahul Dec 29 '22 at 07:03
  • If you take an integral of $|f|$ on $(0,+\infty)$, you get $+\infty$ (from the harmonic series). – mowzorn Dec 29 '22 at 07:05
  • When we say a function $f$ is integrable on $E\subset\mathbb{R}$, we mean $\int_E |f(x)|dx < \infty$. – Riemann Dec 29 '22 at 07:10
  • But for the function sin(x)/x, its integration over $(0,\infty)$ is finite, doesn't it mean that it is integrable over this interval? (Its modulus has no finite integrataion.) – Maths Rahul Dec 30 '22 at 06:24
  • Actually this is a neat example. Riemann integral is defined on bounded intervals, and in that case, as we know, Riemann integrable functions are Lebesgue integrable. However, when we use improper Riemann integral, as you did over $(0,\infty)$, then we might lose that property. And $\sin x/x$ is a standard example of that. Edit: there is a nice discussion of this topic here, if you would like more insight https://math.stackexchange.com/questions/2293902/functions-that-are-riemann-integrable-but-not-lebesgue-integrable – mowzorn Dec 30 '22 at 23:16

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