I think I can learn something from the joke below.
I found this meme online $$\sum_{i=1}^{n}i=\sum_{i=1}^{\infty}i-\sum_{i=n+1}^{\infty}i\\ =\sum_{i=1}^{\infty}i-\sum_{i=1}^{\infty}(i+n)\\ =\sum_{i=1}^{\infty}(-n)=-\infty$$
(I reported exactly the equation as I found it even if it is poorly written: $i$ is the sum index and not anything else).
Some comments stated that the error is in the second term of the second line, some others said the problem is in the first step.
What is wrong and, primarily, why?
Summarizing the facts that are mentioned in the comments under the question:
The sum $\sum_{1}^{\infty}i$ is divergent.
The sum $\sum_{n+1}^{\infty}i$ is divergent.
The difference of two divergent sums is meaningless. Specifically, you cannot say that $$\sum_{1}^{\infty}i - \sum_{n+1}^{\infty}i$$ is equal to anything, let alone say it is equal to $\sum_{1}^{n}i$. So the joke is that the very first equation is already nonsense.