Whers d fallacy??

Let,

\mathcal{S}=a+ar+ar^2+\cdots+ar^n=a+r\mathcal{S}-ar^{n+1}

If 0<|r|<1,

\lim_{n\rightarrow \infty}ar^{n+1}=0

Hence if n→∞,

\mathcal{S}=a+r\mathcal{S}

Therefore,this method can be applied iff the series is converging.The last term of the above series is either 1 or -1 and is not negligible.(i.e \lim_{n\rightarrow \infty}ar^{n+1}\neq0)

You can't write S=1-S

If we try to apply this trick on divergent or oscillating series(like the above one) we end up with weird results.As another example try applying this trick on a divergent series like,

\mathcal{S}=1+2+2^2+\cdots

You'll find that S is negative!