Background: I am subscribed to a math-based YouTube channel. The channel in question recently did a video on Numberphile's goof of saying that 1 2 3 4...=-1/12, and the steps needed to correct the issue. About halfway in the video, the presenter posed a unique convergence problem based on the Numberphile video:

1 0-1 0 1 0...

In the comments, I came up with this solution:

if S=1 0-1 0 1 0...

I supersum for partial sums to get...

1 1 0 0 1 1 0 0

I supersum it again to get the following:

1 1 (2/3) (1/2) (3/5) (4/7) (1/2)

Surprisingly, the channel commented that I was correct in my work and that the last series is convergent. They also challenged me to make the first series supersum to some other value by distributing the 0s in a slightly different way.

It is based on this challenge that I seek help as someone who has no background in calculus, I think that I have a working theory on the situation:

Can 1-1 1-1... Converge to 0 if 0 is added every n^th position?

I think that the above assumption is correct, but I don't know anything about series or calculus to prove it.