I was computing an integral, and in particular the integral is
$$\int_0^{+\infty} \frac{|\sin(2x)|}{x}\ \text{d}x$$
(which by the way is infinite, I forgot to say before).
I really don't know why I decided to use Mathematica to get a "fast" confirm of that but...
Besides the fact it wasn't really fast at all, the result that Mathematica gave me is the following (by the way: I'm talking about the serious pc version of Mathematica, not Wolfram Alpha online):
$$\sum _{K[1]=1}^{\infty } (\text{Si}(2 \pi K[1]+\pi )-\text{Si}(2 \pi K[1]))+\sum _{K[1]=1}^{\infty } (-\text{Si}(2 \pi K[1])-\text{Si}(\pi -2 \pi K[1]))+\text{Si}(\pi )$$
I did not really understand what $K[1]$ is, here, because for one side it seems like a pure index of a sum, like a normal $j$ from $1$ to $\infty$.
On the there side, I found a specific function called $K(n)$ in wolfram documentation, referring to
$$K(n) = 1^1\cdot 2^2\cdot 3^3 \ldots (n-1)^{n-1} = H(n-1)$$
Where $H$ is called Hyperfactorial.
I am really confused about that notation, so could anybody explain me better this ambiguous result?
If it had to be a general sum index, why not to write it as $k$ from $1$ to $\infty$ and stop?
And if it had to be a sum over the argument of $K$ function, why not writing $K(n)$ for $n = 1$ to $\infty$?
Thanks!
Consider $$I=\int_0^{+\infty} \frac{|\sin(2x)|}{x}\,dx=\int_0^{+\infty} \frac{|\sin(y)|}{y}\,dy=\sum_{k=0}^\infty\int_{k \pi}^{(k+1)\pi} \frac{|\sin(y)|}{y}\,dy$$ $$I=\sum_{k=0}^\infty\int_{2k \pi}^{(2k+1)\pi} \frac{\sin(y)}{y}\,dy-\sum_{k=0}^\infty\int_{(2k+1) \pi}^{(2k+2)\pi} \frac{\sin(y)}{y}\,dy$$ $$I=\sum_{k=0}^\infty\left(\text{Si}( (2 k+1) \pi )-\text{Si}(2k \pi )\right)-\sum_{k=0}^\infty\left(\text{Si}(2 (k+1) \pi )-\text{Si}(2 (k+1)\pi )\right)$$ $$I=\sum_{k=0}^\infty\left(2 \text{Si}((2k+1) \pi )-\text{Si}(2 k \pi )-\text{Si}(2 (k+1) \pi )\right)$$