While I understand the point you are trying to make, your assertion is flawed in a simple way: you assume that "converges to" and "equals" are the same thing, when discussing its value. Grandi's series doesn't converge; however, it can be considered equal to 1/2. This can be observed by seeing that its value is 1/2 using multiple different techniques, including continuation, Cesaro summation, and algebraic evaluation (as done by the person you were replying to).

Your argument in the article itself for why Z=1/2 doesn't make sense is flawed, as the series is clearly not absolutely convergent, and thus reordering an infinite number of terms can result in a different value, even where a series can converge (the classical example being the alternating harmonic series). To get "Z+Z=Z", you must rearrange an infinite number of terms, and thus the reasoning cannot work. On the other hand, the reasoning provided by the anonymous poster you replied to requires only a finite insertion, and thus "1-Z=Z" is reasonable.

Of course, the video (and thus Ramanujan) has the same flaw in their reasoning - they're reordering the infinite sums in a way that is inconsistent... but the video was just for lay people, and wasn't meant to be rigorous. More formal techniques agree with the result, and the fact that the same results can also be seen in physics indicates that "1+2+3+...=-1/12" is a meaningful statement, even if it shouldn't be interpreted as "the limit of the partial sums" in any sense - it shows that it isn't just a bit of mathematical sleight-of-hand.

While I understand the point you are trying to make, your assertion is flawed in a simple way: you assume that "converges to" and "equals" are the same thing, when discussing its value. Grandi's series doesn't converge; however, it can be considered equal to 1/2. This can be observed by seeing that its value is 1/2 using multiple different techniques, including continuation, Cesaro summation, and algebraic evaluation (as done by the person you were replying to).

Your argument in the article itself for why Z=1/2 doesn't make sense is flawed, as the series is clearly not absolutely convergent, and thus reordering an infinite number of terms can result in a different value, even where a series can converge (the classical example being the alternating harmonic series). To get "Z+Z=Z", you must rearrange an infinite number of terms, and thus the reasoning cannot work. On the other hand, the reasoning provided by the anonymous poster you replied to requires only a finite insertion, and thus "1-Z=Z" is reasonable.

Of course, the video (and thus Ramanujan) has the same flaw in their reasoning - they're reordering the infinite sums in a way that is inconsistent... but the video was just for lay people, and wasn't meant to be rigorous. More formal techniques agree with the result, and the fact that the same results can also be seen in physics indicates that "1+2+3+...=-1/12" is a meaningful statement, even if it shouldn't be interpreted as "the limit of the partial sums" in any sense - it shows that it isn't just a bit of mathematical sleight-of-hand.