The harmonic Series sequence. Well! I was going through harmonic series from mathworld.worlfram I found harmonic numbers are really tough to calculate I was scribbling and wrote this thing
$$\sum_{k = 1}^{x}H_k= \sum_{k =1}^{x}\left(\sum_{u = 1}^{k}\frac 1{u}\right)$$
How can I simplify these highly complicated things Howe can we find this thing
At the first, we can't find $H_k$ itself and series over series making it a little hard
I wonder how can we simplify this thing?
It's just my curiosity....

Well! then I liked this curiosity
I guess by solving you mean just to find only one harmonic number(say: $H_p)$ and finding the above $\sum\sum\frac 1{k}$
This is possible and I solved that by finding a pattern that only the last terms of each harmonic number are making previously calculated harmonic numbers different with only $\frac 1{k-1}$
But for the sake proof of my simplified result, I can provide an easy way
$$\begin{align*}
S 
&=\sum_{k =1}^{x}H_k\\
&=\sum_{k =1}^{x}\sum_{n =1}^{k}\frac 1{n}\\
&=\sum_{k =1}^{x}\frac 1{n}\sum_{k =n}^{x}1\\
& = \sum_{k =1}^{x}\frac 1{n}(x-n+1)\\
& =\sum_{k =1}^{x}\frac {(x+1)-n}{n}\\
& =(x+1)\sum_{k =1}^{x}\frac 1{n} - \sum_{k =1}^{x}1\\
& = (x+1)H_x - x
\end{align*}$$

Here, SH() & test() is our result expected which are equal.
