<s>[INST] Addition of fractions repetition and convergence 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.
 [/INST] $$ 
if\\n\neq 1 \\ s=\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\frac{1}{n^4}+...\\multiply\\ s \\by \\n \\\frac{1}{n}s=\frac{1}{n^2}+\frac{1}{n^3}+\frac{1}{n^4}+\frac{1}{n^5}...\\now\\s-\frac{1}{n}s =\frac{1}{n}+(\frac{1}{n^2}-\frac{1}{n^2})+(\frac{1}{n^3}-\frac{1}{n^3})+(\frac{1}{n^4}-\frac{1}{n^4})+(\frac{1}{n^5}-\frac{1}{n^5})+...\\so \\s-\frac{1}{n}s =\frac{1}{n}
 $$
$$
s(\frac{n-1}{n})=\frac{1}{n}
$$
$$
s=\frac{1}{n-1}
$$
</s>