You can check at

https://www.cut-the-knot.org/arithmetic/999999.shtml

that the same "sum" also equals 1/2.

>as far as I can see I've not done anything wrong,

Why do you think it is possible to regroup an infinite series?

> but my
>teacher says you can't regroup infinitely long sums in this
>way, why not?

Close your eyes, take a deep breath and try to change your mind set. The reason you can't regroup infinite series, in general, is exactly because sometimes you get results like 0 = 1.

You start with an assumption that a number could be somehow associated with 1-1+1-... and that usual arithmetic rules apply to infinite series. The identity 0 = 1 that you obtained proves your assumptions wrong.

Some sums, however, could be associated with unique numbers, and others allow regrouping of terms without changing the related number.
The former are known as convergent series, the latter is true for absolutely convergent series.

Look at the partial sums, S_n:
S_1 = 1
S_2 = 1-1 = 0
S_3 = 1-1+1 = 1
S_4 = 1-1+1-1 = 0
...
S_n = 1/2 (1 - (-1)ⁿ)

The "..." at the end of your sum is taken to mean "the limit of S_n as n gets very large" (where 'limit' can be defined formally if required).

But if S_n doesn't get very close to a particular limit (and you can use infinity as the limit if you are careful about what that means) then the series is meaningless.

Let's say you were standing on a graph of "1/x", and looked from the origin towards infinity - if you looked far enough away the graph would "equal" zero; but if you looked along a graph of S_n against n, it would not equal any single value however far away you looked.

In this case the series doesn't make any sense, so if you assume it does, you have introduced a contradiction into the system and can prove anything.

Rich