It is true that the number of ones shown(!) changes from 8 to seven, but the "..." at the end means that the sum goes on to infinity, i.e. there are infinity "1"s and infinity "-1"s, regardless of how many are written down. You can continue adding in "1"s and "-1"s by hand if you want, but your arm will get tired before infinity, so best to use the "..."!

The point here is that if you pair the terms "1" and "-1" in different parts of the sum, you seem to get a different answer, which is only true because the sum goes to infinity. Going to infinity means that you can just keep adding paired terms that equal 0 in this way, so there seems to be a problem with infinity and making the sum make sense. It is certainly true that for any finite sum there is absolutely no problem with how we group the terms, the answer is the same no matter what. Hope this helps.

It is true that the number of ones shown(!) changes from 8 to seven, but the "..." at the end means that the sum goes on to infinity, i.e. there are infinity "1"s and infinity "-1"s, regardless of how many are written down. You can continue adding in "1"s and "-1"s by hand if you want, but your arm will get tired before infinity, so best to use the "..."!

The point here is that if you pair the terms "1" and "-1" in different parts of the sum, you seem to get a different answer, which is only true because the sum goes to infinity. Going to infinity means that you can just keep adding paired terms that equal 0 in this way, so there seems to be a problem with infinity and making the sum make sense. It is certainly true that for any finite sum there is absolutely no problem with how we group the terms, the answer is the same no matter what. Hope this helps.