Alternative Definitions of Convergence for Sequences In particular, my question centers around, and stemmed from, consideration of the geometric series. According to the usual definition of convergence, the closed form for the geometric series only "makes sense" when the magnitude of the argument is less than unity. But, certain arguments still "make sense", despite their being outside the radius of convergence. 
$$
\frac{1}{1+1} = 1 - 1 + 1 - 1 + ... = \frac{1}{2}
$$
$$
\frac{-1}{1+1} = -1 + 1 - 1 + 1 - ... = -\frac{1}{2}
$$
$$
\frac{1}{1-1} = 1 + 1 + 1 + 1 + ... = \infty
$$
Those are the ones that make sense intuitively. However, most curious are the ones that "don't make sense": 
$$
\frac{1}{1+2} = 1 - 2 + 4 - 8 - ... = \frac{1}{3}
$$
$$
\frac{1}{1+n} = 1 - n + n^2 - n^3 - ... 
$$
I've tried to formulate different senses of convergence in which these relations hold generally (the usual ones, the intuitive ones, and the curious ones), but didn't get anywhere. Is anyone aware of any other definitions that might be relevant here? I would be grateful. I think I remember once hearing that Euler was also interested in such formulae but I'm not sure. 
-Andrew.
 A: You can read the first chapter of the book "adventures in formalism". It contains a very detailed account of the history of notions of convergence related to the series you describe. 
You might like to know that in the early days of calculus you would have been in good company trying to figure out such strange 'results' obtained by plugging in values for variables without worrying about convergence, and being surprised that sometimes you get sensible things, sometimes you get nonsense, and sometimes you get things that seem oddly strange, and thus think they must have some meaning.
Today though, there is no issue here and the situation is well understood. Certain functions may be seen as generating functions for certain series or sequences, but only inside the radius of convergence. There is also a lot known about the behaviour of power series on the edge of the circle of convergence. 
A: The wikipedia article on divergent series has a pretty good high-level overview of various summation methods. That is, ways to get meaningful values as "sums" of divergent series.
One intuitive summation method is Abel summation. Given the series $\sum_n a_n$, you look at $\lim_{r\to 1^-} \sum_n a_n r^n$.  
For example, for your first series, $\sum_{n=0}^\infty (-1)^n$, you get
$$
\lim_{r\to 1^-} \sum_{n=0}^\infty (-1)^n  r^n = \lim_{r\to 1^-} \frac 1 {1+r} = \frac 1 2.
$$
If you want to go further, Stein and Shakarchi's introductory book on Fourier analysis uses Cesaro and Abel summation to deal with the divergent sum $\sum_n e^{in\theta}$.  Abel summation gives you the Poisson kernel, which is used to solve the heat equation on the disk.
