Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.
And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.
If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.
But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.
Which one is right? Or are these propositions depends on what we define?
Formally speaking, the series is divergent.
The sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. We see that the sequence of partial sums of the series, known as Grandi's series is $1, 0, 1, 0, …$, which does not approach any number (although has two accumulation points at $0, 1$). Therefore, we may conclude that Grandi's series (http://en.wikipedia.org/wiki/Grandi's_series) is divergent.