# Simplify the following sum $sum_{i=1}^nfrac1{n-(i-1)}$

Mathematics Asked by Wild on December 3, 2020

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:

$$sum_{i=1}^nfrac1{n-(i-1)}$$

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)

$$sum_{i=1}^nfrac1{n-(i-1)} =sum_{i=1}^nfrac1{n} – sum_{i=1}^nfrac1{i} – sum_{i=1}^n1 = 1 – H_n – n$$

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 $$j=nto1$$ And it is given by $$sum_{j=1}^nfrac 1 j$$

Correct answer by NL1992 on December 3, 2020

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