I got stuck in a problem in the middle of my calculations of integrals and sums.
$$\lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty f(n-\epsilon)-f(n+\epsilon)=0$$
where $f$ is continuous on all of the values in the summation
I am not sure for the general case of any $f$, but I needed verification for $f(n) = -\frac{1}{n}$ will be gratefull for an explanation for the any function
First I tried to verify for no sum and here what I got;
$$\forall_{n \in \mathbb{R}} \;\lim_{\epsilon \rightarrow 0} \frac{1}{n+\epsilon} - \frac{1}{n-\epsilon} = \frac{1}{n} - \frac{1}{n} = 0$$
$$\lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty \frac{1}{n+\epsilon} - \frac{1}{n - \epsilon}= \lim_{\epsilon \rightarrow 0} \sum_{n=1}^\infty\frac{-2\epsilon}{(n+\epsilon)(n-\epsilon)} = 0$$
Is this correct? And what about any other function