Sum of a divergent series depends on the way how one performs the summation

Question

Can some one explain the following sentence regarding summation of divergent series?

$\text{Sum of a divergent series depends on the way how one performs the summation}.$

What does mean it?

Can someone explain it by giving an example?

Thanks

Answer 1

Try looking at $$\sum_{n=0}^\infty (-1)^n;$$ depending on how one groups the terms, one has $0$ or $1$:

$$ 1 + (-1 + 1) + (-1 + 1) \to 1\\ (1 - 1) + (1 -1) \cdots \to 0 .$$

Answer 2

I believe it is possible that the theorem you are alluding to concerns, not divergent series, but rather conditionally convergent series.

If you insist on 'divergent', meaning 'non-convergent', then there are different ways a series may fail to converge; for some the statement given holds, for others not. Here the comment by @Somos is pertinent.