Extremely simple question, but one I am struggling with (I haven't taken a math class for a couple years so this may be very easy). I just need to simplify the following sum, but can't seem to figure out how:
$\sum_{i=1}^n\frac1{n-(i-1)}$
I am really struggling to find where to start, since it seems you would need to break the sum into parts to simplify, but I don't think you can. My best guess would be doing something like below, but I don't think you can break the fraction into parts like this (but can't think of any other way to solve)
$\sum_{i=1}^n\frac1{n-(i-1)} =\sum_{i=1}^n\frac1{n} - \sum_{i=1}^n\frac1{i} - \sum_{i=1}^n1 = 1 - H_n - n$
I have also heard from a semi-reliable force that this equals $H_n$ but not sure if this is right/how it was reached.
Set $j=n-(i-1)$, then substituting, the first sum is over $j=n\to1$ And it is given by $\sum_{j=1}^n\frac 1 j$