The series $1-1+1-1+1-1+1-\dots$ diverges.

Solution. The odd partial sums are all $1$; whereas the even partial sums are all $0$; so the series diverges.

Here $a=1$ and $r=-1$ and $a/(1-r)=1/2$, but we cannot use the geometric sum formula since the condition $-1<r<1$ does not hold.