1+2+3+4+...

"I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."

Ramanujan to Hardy



In my opinion it is easier to see that s = 1 + 2 + 4 + 8 + ... = -1, by writing
s = 1 + 2*( 1 + 2 + 4 + ... ) = 1 + 2s and from s = 1 + 2s it follows that s = -1.