OMG !! What Have YOU Done?????
Z+Z= the original sum i.e. Z
so you write 2Z=Z
so putting Z=0.5, we get 0.5=1
but why stop here?
It's an infinite sum so at my will I can write (Z+Z = Z+Z) or (Z+Z = Z+Z+Z+.........+Z)
so, Z=Z or Z=nZ
so whatever Z is, Z can be anything else....it's nonsense. RIGHT??
Don't stop. It's not like counting bricks. Whenever you put an end to it, its a divergent series and for that don't assign any value for GOD's sake!
BUT, try to think it this way. The sum is indeed INFINITE. That is the only truth. It doesn't end. But that infinite is a pile of junk. There is a bit of finite part in that infinity. Whenever you stop and calculate, each time it blows up. But there is this (-1/12) part which only accumulates if you allow it to go on forever. And yes we physicists use it in quantum fields and all sort of places. Each time we replace that infinite sum with this very value, we confirm the result! Don't teach things you don't fully understand.