Can we assign a value to the subtraction of two divergent series summing over equivalent terms?

Let's say we have $S=\sum_{n=0}^\infty{a_n} - \sum_{n=0}^\infty{a_n}$. Can we say that $S=0$ even if $\sum_{n=0}^\infty{a_n}$ diverges?

Maybe one potential argument to say is so is considering partial sums, as follows:

$S=\lim_{N \to \infty} \sum_{n=0}^N{a_n} - \sum_{n=0}^N{a_n}=\lim_{N \to \infty} \sum_{n=0}^N{(a_n-a_n)}=\lim_{N \to \infty} \sum_{n=0}^N{0}=0$

But is this a valid reasoning?

I would appreciate any clarification on the matter.

Solution 1:

The expression $\sum_{n=0}^\infty a_n - \sum_{n=0}^\infty a_n$ is $0$ if $\sum_{n=0}^\infty a_n$ converges, and otherwise it is not defined. You cannot subtract two things that are not defined: in what set should this operation take place?

Try to always have a clear idea of what objects you are considering and what operations you are allowed to perform on them. What should $S$ be, a number? The limit of a series? Which one? You can only deduce clear properties if you start from clear definitions.