Proof why $\frac{1}{n^x}+\frac{1}{(n+1)^x}=\frac{H_{{n+1,x}}}{n^xH_{n,x}}+\frac{H_{{n-1,x}}}{(n+1)^xH_{n,x}}$ I found this in a not very straightforward way, and it seems like a rather strange-looking identity, but there is probably a simple proof.
$$\frac{1}{n^x}+\frac{1}{(n+1)^x}=\frac{H_{{n+1,x}}}{n^xH_{n,x}}+\frac{H_{{n-1,x}}}{(n+1)^xH_{n,x}}$$
Where $H_{n,x}$ is the Generalized Harmonic number.

Using the simple computation
$$ \frac{H_{n,x}}{n^x}+\frac{H_{n,x}}{(n+1)^x}=\frac{H_{n+1,x}-(n+1)^{-x}}{n^x}+\frac{H_{n-1,x}+n^{-x}}{(n+1)^x}=\frac{H_{n+1,x}}{n^x}+\frac{H_{n-1,x}}{(n+1)^x}, $$
 the result follows immediately.
