Right before They speak on Converg n Diverg Series

We know on studying series, we have some tight rules. One of it is the convergent of a define series. a Series with equal to S have to prove its convergent or not. Or its should convergent on its tail.

But right before it, we don't use it convergent to determine the sum of series. Its a nice sample below n how mathematicians confuse before.

Define a series below

1 - 1 + 1 - 1 + 1 - ...

So, when we first meet this situation, we will do an association that the series will become like this.

(1 - 1) + (1 - 1) + ...

Since for each (1-1) equal to zero, we know that the whole series is equal to Zero too..
But now think about doing same with another different point of views, and we get the new series below

1 + (-1 + 1) + (-1 + 1) + ...

Since for each (-1 + 1) is equal to zero, we will find it left a one on that series. Then we conclude that the whole series is equal to 1.

That make a paradox. How could an identical series have 2 solution for S, that S is sums of series? And we know that the procedure to get the 2 of solution is comfirmed as legal. And you will get more confuse (before the convergent spoked) with this method. We use whole series as a symbol we called S, as below.

S = 1 - 1 + 1 - 1 + ...

and We do some distribution n association below

S = 1 - (1 - 1 + 1 - 1 + ... )

since we know that S is the series itself,

S = 1 - S

then we add an S to both side and we get

2S = 1

So we conclude that S, that is whole series itself, is equal to an half. The solution will lead us to 0 = 1 = 1/2, that is insane, so the consensus make the convergent of the series as a factor to satisfy before we find the solution.