TITLE: sum of the series?
QUESTION [5 upvotes]: let sum of the series 
$$S=1-1+1-1+1-1+1-1+1-1........$$
$S=\frac{1}{2}$
my question is if there are even number of terms then the sum is $0$ 
and if the number of terms are odd then sum is $1$.
but we don't know whether its odd or even because the sequence goes to infinity.
Is the answer derived by using the probability of getting either $0$ or $1$

I am making an edit to this, can it done by using the 
where $i $ is 4th root of unity

$$S=i+i^2+i^3+i^4+i^5.....$$

$$S=i-1-i+1+i-1-i+1.....$$
then $1$ can be summed up using infinite sum of Gp
which is $$S=\frac{a}{1-r}=\frac{i}{1-i}$$
by simplifying we get $$S=\frac{i}{1-i}\frac{1+i}{1+i}=\frac{i-1}{2}$$
and take the real part of $S=Re \big[ \frac{i-1}{2}\big]=-\frac{1}{2}$
is it a wrong way to solve this problem ???

REPLY [4 votes]: Look this(Terence Tao-Analysis I):]1