Grandi's Series; tends to $1/2$, but why is this considered a valid sum?

Consider power series $$ S(x) = \sum_{n=0}^{\infty} (-1)^n x^n, \qquad x\in [0;1). $$ It is geometric series: $$ \sum_{n=0}^{\infty} (-x)^n = \frac{1}{1-(-x)} = \frac{1}{1+x}. $$

So, $$ S(x)=\frac{1}{1+x}, \qquad x\in[0,1). $$

$S(x)$ is continuous and bounded on $[0;1)$. So, we can find limit: $$ S = \lim_{x\to 1} S(x) = \frac{1}{2}. $$

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See Abel summation for better understanding.

The series does not converge it is not mathematically valid if we look at the epsilon definition of convergence. The result is useful in physics and is used there.