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Assume we have the sum $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ why is this equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?

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    $\begingroup$ Why is $$\frac55+\frac54+\frac53+\frac52+\frac51$$ equal to $$5\left(\frac11+\frac12+\frac13+\frac14+\frac15\right)?$$ $\endgroup$ – Lord Shark the Unknown Apr 12 at 17:11
  • $\begingroup$ One is equal to $n(\frac1n + \frac1{n-1} + \cdots + \frac11)$ whereas the other to $n(\frac11 + \frac12 + \cdots + \frac1n)$ $\endgroup$ – Alonso Delfín Apr 12 at 17:16
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$$\sum_{i = 1}^n \frac{n}{n-i+1}=n\sum_{i = 1}^n \frac{1}{n-i+1}=n \left(\frac{1}{n}+\frac{1}{n-1}+...+\frac{1}{1}\right)$$

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The first comment above gives an excellent reason, but you can formalize it as follows: substitute $\;k:=n-i+1\;$ , then this is an ascending function of $\;n\;$ (differentiate, for example...or directly), and we get that

$$i=1\implies k=n\;,\;\;i=n\implies k=1$$

so we in fact get

$$\sum_{i=1}^n(n-i+1)=\sum_{k=1}^nk$$

and in the original question:

$$\sum_{i=1}^n\frac n{n-i+1}=n\sum_{i=1}^n\frac1{n-i+1}=n\sum_{k=1}\frac1k$$

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If you simply open the summation you see that:

$$\sum_{i=1}^n\frac n{n-i+1}=\frac n{n}+\frac n{n-1}+\frac n{n-2}+\cdots+\frac n{1}$$$$=n\bigg(\frac 1{n}+\frac 1{n-1}+\frac 1{n-2}+\cdots+\frac 1{1}\bigg)$$ $$=n\bigg(\frac 1{1}+\cdots+\frac 1{n-2}+\frac 1{n-1}+\frac 1{n}\bigg)$$ $$=n\sum_{i=1}^n\frac 1{i}$$

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