Divergent Series Equality?

Is the square of the alternating sum of the sequence of ones equal to positive integers? Well, they're both divergent series, so the interpretation doesn't really make sense. But if we treat them like convergent series and apply theorems that we aren't supposed to, we get a visual proof that these are the same. To square an infinite series, we multiply each term by every other term. We can imagine doing this in an infinite square array. The entry in row i, column j is the product of the i and the j sum ends from the series. We let positive 1 be a shaded cell and negative 1 be an unshaded cell. So the resulting rectangular array looks like an infinite checkerboard. If these were convergent, we would add along anti-diagonals, which corresponds to shifting the I column down I units and adding along the now finite rows. As we move down the triangular staircase, row one represents positive 1, row two represents -2, row 3 is 3, row 4 is4, and so on. Continuing indefinitely, this diagram represents the alternating sum of positive integers. So if we treat them like convergent series, we see that the square of the alternating sum of the sequence of ones is equal to the alternating sum of positive integers. What other bogus results can you get from this process?