TITLE: Addition of fractions repetition and convergence
QUESTION [1 upvotes]: Is this a new mathematical concept?  
$$
\frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3} \cdots = \frac{1}{n-1}
$$
If it isn't then what is this called?
I haven't been able to find anything like this anywhere.

REPLY [0 votes]: Fix $r>0$ and consider
$$
1+r+r^2+r^3+\dots+r^k
$$
Multiplying by $(1-r)$ you obtain
$$
(1+r+r^2+r^3+\dots+r^k)(1-r)=\\
(1-r)+(r-r^2)+(r^2-r^3)+\dots+(r^{k-1}-r^k)+(r^k-r^{k+1})=\\
1-r^{k+1}\;\;.
$$
Hence you have the following identity (valid, as you can see, for $r\neq1$):
$$
1+r+r^2+r^3+\dots+r^k=\frac{1-r^{k+1}}{1-r}\;\;.
$$
Then you can pass to the limit $k\to+\infty$, obtaining a finite quantity iff $|r|<1$.
This quantity clearly is
$$
\frac1{1-r}
$$
i.e.
$$
1+r+r^2+r^3+\dots+r^k+\dots=\frac1{1-r}\;\;.
$$
Now, in order to obtain your result just pose $r=1/n$, for a fixed $n$; I guess it's a natural number, hence if $n\ge2$, clerarly $r=1/n<1$ and the above formulas are valid and easily yeld to your result.
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