how would one prove that claim? it's not obvious to me. it seems like the specific conditions under which the series of terms in the total expectation has a well defined sum is important, but I didn't fully grasp those conditions

For example if instead of halving the probability of each value as he does in the video we take 1/20 then we get the expected value as the sum of 9/20^2 - 9/20^2 + 90/20^3 - 9/20^3 + … Here the series is absolutely convergent since the positive terms sum to 9/10 and the negative terms sum to 9/10.