Grandi's series is defined as:
$$\sum_{n=0}^{\infty} (-1)^n = 1 - 1+1-1+\cdots$$
By plainly looking at this series it seems like the value of it is either $1$ or $0$ by doing the following groupings:
$$(1-1)+(1-1)+(1-1)+\cdots=0+0+0+\cdots=0$$
OR
$$1+(-1+1)+(-1+1)+\cdots=1+0+0+\cdots=1$$
However, if we say
$$A = 1 - 1+1-1+\cdots$$
Then
$$1-A = 1-(1 - 1+1-1+\cdots)=1 - 1+1-1+\cdots = A$$
and thus $A = 1/2$
We know that $\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots$ which, when $x=1$, evaluates the sum as $1/2$ once again.
One final proof that the series converges to $1/2$ is as follows (similar to the above proof): $$1/2=\frac{1}{1+1} = 1-\frac{1}{1+1} = 1-\left(1-\frac{1}{1+1}\right)=1-\left(1-\left(1-\left(1-\left(\cdots\left(1-\frac{1}{1+1}\right)\right)\right)\right)\right) = 1-1+1-1+\cdots$$
This series apparently confused many of the great minds in math. Many, many more methods are known that also show that this sum is equal to $1/2$.
Why do many seemingly divergent sums such as this one seem to converge? Some other examples that can be evaluated include $1 − 2 + 3 − 4 + \cdots = -1/4$, $1 − 2 + 4 − 8+ \cdots = 1/3$, $1 + 1 + 1 + 1 +\cdots=-1/2$, $1 + 2 + 3 + 4 +\cdots=-1/12$, etc.
See http://en.wikipedia.org/wiki/Grandi's_series for more information about this series.