On that first Numberphile video, I'm confused why he shifts the S2+S2 second sequence to the right, but doesn't do something similar when he subtracts S-S2. Everything about this screams wrong to me. I'll have to watch that second video whenever I have a bit more time.

You're absolutely right for it to scream wrong to you. The manipulations are done cavalierly in that divergent series are treated as if they are numbers. It even starts with a piece of nonsense:

**let ***c* or *x* or *S*= 1+2+3+4+... (depending on which part of the internet you are).

This equation has no meaning. The right hand side, whilst easy to understand, is undefined mathematically so to say it is equal to something is preposterous. Any algebraic jiggery-pokery after that is just flim-flam and sophistry.

We might as well claim that 1 = 3 and from that find that 2 = 6 and 2 = 4.

In fact, I'm going to have a bash at this infinity business. I claim that

2 + 5 + 8 +.....+ (3*n*-1)+...=0

A prize will be awarded to the first person who works out how I did this.

The sad part is that there are some beautiful results involving infinity in mathematics which are relatively easily explained to anybody with sufficient curiosity.