Extreme care is needed when dealing with infinite sums. From what I can tell, it seems that confusion arises when people assume they are allowed to manipulate infinite sums in the same way as they are used to doing with the finite sums they are accustomed to dealing with. Assuming infinite sums behave anything like finite sums will lead to problems. Reordering a non-convergent infinite sum will often produce different answers - in many cases you can get any result you desire, but it's clearly incorrect. Proceeding term-by-term to calculate the partial sums. What does it even mean to sum an infinite number of terms when the sum is non-convergent? It's unsafe to assume that what works for finite sums must apply to infinite sums as well.
Regarding -1 + -1 ... (Grandi's Series), although it doesn't have a limit, it is Cesaro summable and as a cesaro sum of 1/2. This can be easily proved in lots of ways. This is what Numberphile were talking about no doubt.