Summing an infinite series paradox

0 = 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4                                                     L1
The above is fine, but what about the infinite version:

0 = 1 - 1 + 2 - 2 + 3 - 3 + 4 - 4 + ...                                              L2
There is nothing wrong with writing out infinitely many terms.

Yet we can re-arrange it to give
0 = 1 + ( -1 + 2 ) + ( -2 + 3 ) + ( -3 + 4 ) + ( -4 + 5 ) + ...             L3
   = 1 + 1 + 1 + 1 + ...
   = infinity

What is the way out of this?

L2 is invalid as the RHS cannot be summed. The RHS of L2 is an oscillating series ie it does not have a sum and hence cannot be zero. So we need to forbid the infinitely repeated expansion of 0 to be 1 - 1, 2 - 2, 3 - 3 etc as written above (ie without brackets).

We could approach it another way:
0 = 1 - 1
0 = 2 - 2
0 = 3 - 3
and so on.

Then we can add together this infinite number of equations to get L2. What could be wrong with that?

Answer: Adding these up does not give us L2 but
0 = ( 1 - 1 ) + ( 2 - 2 ) + ( 3 - 3 ) + ( 4 - 4 ) + ...                            L4

This is a conditionally convergent series from which we cannot validly remove the brackets.

Note that a subtle change occurred when we added all the equations together. Each of the original equations had an even number of terms on the RHS, whereas their sum does not have a specific number of terms ie the number is neither even nor odd.