Nested Sigma Noation

Actually, the formula is:

Do be careful about jumping straight to a/(1-r) for something that looks like a geometric series.

Otherwise

1 +2 +4 +8 +...

Adds up to

1/(1-2) = -1

Pretty sure it doesn't add up to negative one.

the whole being able to add up to a/1-r only holds for |r| <1. So using it for when your n =1 (hence r=1) isn't valid. Not to say the rest of what you're saying is wrong. The sum of 1's definitely diverges... you just shouldn't do it by changing a geometric series to a/1-r when |r| isn't always <1.

Granted, it's quite obvious it only fails when your n=1. But I'd just like to point out... be careful.