You are actually allowed to add up infinite series like this.
Only that the infinite series have to be convergent, or else you get little of value. The series in your example oscillates forever (and the oscillation distance remains constant), therefore it diverges.
Take the infinite series 1 + 0 + 0 + 0 + ... and add it like you did:
1 + 0 + 0 + 0 + 0 + ... ___ 1 + 0 + 0 + 0 + ...
And you just get 1 + 1 + 0 + 0 + 0 + ... which is just 2 * (1 + 0 + 0 + 0 + ...)
You are actually allowed to add up infinite series like this.
Only that the infinite series have to be convergent, or else you get little of value. The series in your example oscillates forever (and the oscillation distance remains constant), therefore it diverges.
Take the infinite series
1 + 0 + 0 + 0 + ...and add it like you did:1 + 0 + 0 + 0 + 0 + ...___ 1 + 0 + 0 + 0 + ...And you just get
1 + 1 + 0 + 0 + 0 + ...which is just2 * (1 + 0 + 0 + 0 + ...)