I need help converting this summation to an integral

$$\sum_{i=1}^\infty 1+\frac{\vert(n-im)\vert}{(n-im)}$$

I keep trying but get stuck so any help is appreciated.


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The fraction is $+1$ or $-1$ depending on whether $i \lt \frac nm$ or $i \gt \frac nm$. If $m$ divides $n$ you have a term the divides by $0$ so the sum is undefined. Assuming you don't fall into the undefined trap, the summand is just $2$ as long as $i \lt \frac nm$, and $0$ after, so the sum is just $2\lfloor \frac nm \rfloor$. To convert it to an integral the function is just $2$ from $0$ to $\lfloor \frac nm \rfloor$ and $0$ after.


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