Yes and no.

In this case we are asking what the end result of an alternating sequence will end at, and the intuitive answer is what we take: 1/2.

That said, this isn't math from High School Algebra, or even College Algebra. We are tending into things that are wrapped up in arguments of Schrödingers cat where something takes on two values, seemingly at the same time. Until we open the box, or in this case cut off the series, we do NOT know the value. This is key.

IF we cut the series down, we are now truly in the land of 'equals' and can state that 1+2+3+4+5=15! However, when we look at 1+2+3+4+5+... when we do that, we have stepped into another thought process since we are tending to infinity and infinity can never be defined as a simple number, even if large, because you just at one to it and you have a new infinity, to which you add one and so on... thus, we choose a symbol meaning "big number, bigger than any other, what it is we do not know"... worse, we have just proved that there is no such number (cause you can just add one to it!).

Ok, enough hand waving. Yes, they are abusing the equals sign a bit. I took offense when they used the right arrow, ->, which means "tends to" rather than "is equal to" which can also be "can be replaced by." So, I feel your angst at that. Trust me that we leave that version of the equals sign behind when we start working with numbers like infinity.

The kicker here is that physics proves out this sort of strange stuff. I have flown in a Chinook Helicopter... AFTER proving it cannot fly! Imaginary numbers keep popping out and if you don't like them, planes fall out of the sky. If you tolerate them, planes do just fine. Well, planes DO fly, the Chinook is the ugliest helicopter that does amazing amounts of work in the sky (some of the time at least), it DOES indeed fly. So, we have to tolerate these things until we truly understand them.

I think that some day we will have a better way of explaining what the imaginary number means, and what infinity means. Just as we are now better able to explain quantum mechanics (when I was studying it, the guys writing the books understood it but decidedly did NOT know how to write about it).

I don't know if this helps. I hope it does... it does seem a bit strange and a bit like hand waving but, the math is tight.

Yeah, you need to watch the videos and how they arrived at the conclusion.

ALSO, this is not a "new thing".

And not a "thing" that some YouTuber made-up. Numberphiles, more specifically the YouTube channels produced by Brady Haran (http://www.bradyharan.com/), are a bunch of physics, mathematics, chemistry professors - one of them even getting knighted (https://en.wikipedia.org/wiki/Martyn_Poliakoff). And Mathologer has his own mathematical credentials.

Here's the head F'-ery in Wikipedia: [goo.gl] (Had to use Google's URL shortener because the direct link to Wikipedia broke when posting as a comment.)

They're not sure if Euler first found this number or whether it was Ramanujan. Although Euler predates Ramanujan by 100 years, they're unsure if which work specifically stated the sum.

@magicwatch No I am a believer in Brane Theory.