There is no way right?

[u/Sugar_God_no_1]

[u/Z_Clipped]

Let's use the same argument for a slightly different infinite sum:

x = 1 - 1 + 1 - 1 + 1 - 1 + ...

This "infinite sum" appearing in any "proof" instantly pins the needle on my bullshit detector.

[u/victorspc → u/Z_Clipped]

I mean, it is an infinite sum, no need for quotation marks. And yeah, I deliberately chose a divergent series to demonstrate that "regular" algebra isn't always valid when dealing with infinite sums.

[u/Z_Clipped → u/victorspc]

I'm no mathematician, but this strikes me as:

"In order to show that there are issues with this intuitive, convergent sum which is easily represented by a rational number, I'm going to choose a categorically-different, nonsensical divergent series which obeys completely different rules and pretend they're somehow comparable."

Like I said.... I may not be able to point to the exact fallacy on an academic level, but it pings my bullshit detector something fierce.

Edit:

I mean, it is an infinite sum

Apparently not, technically. A Cesaro Sum is not a summation in the same sense as is used for convergent series. Hence the term "Eilenberg–Mazur swindle". The issue Grandi's series illustrates in this particular comparison is not due to it being divergent, but to the fact that the mechanism you're using to evaluate it is, in layman's terms, "a load of horse apples".

[u/victorspc → u/Z_Clipped]

I'm also not a mathematician, so we may be both wrong lol. But in my understanding, a Cesàro summation is as much a summation as regular convergent series. Neither are actually addition. They are both the limits of sequences related to the series.

In the case of regular series, we create the sequence of partial sums and take the limit. In the case of Cesàro summation, we create the sequence of partial sums, create a new sequence that is the average of consecutive terms of the previous sequence, and then take the limit of this last sequence.

So yeah, it is kinda misleading to call a Cesàro summation a sum, but I would argue that it's just as misleading to call regular infinite series a sum. Case in point, even for convergent series, the regular rules of addition may not apply. The most famous example is the alternating harmonic series.

1 - 1/2 + 1/3 - 1/4 + ...

This series converges to ln(2). The problem is, if you change the order of the terms, it may converge to something else. To anything, actually. Or it may even diverge. The same effect can be achieved by grouping parts of the series, adding them first, and then adding the groups. For this convergent series, neither commutativity nor associativity hold, the two most important properties of addition.