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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.

One Answer

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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