Newton's Binomial Theorem with $n<0$

Solution 1:

This is known as Grandi's series. It diverges in the usual sense of convergence (limit of partial sums). However, there are other notions of convergence for infinite series that are more forgiving. For example, using Cesàro summation (limit of arithmetic means), it converges to $\tfrac{1}{2}$.

If this bothers you, you're not alone! But rather than fret, consider this: the fact that your intuition about finite things doesn't extend well to infinite things is not the fault of the infinite things.

Divergent series are weird and wonderful! Remember, that despite speaking of the "sum" of an infinite series, there is no such thing. We can only sum a pair of numbers $a_1+a_2$, then by induction and associativity, we can extend that to a sum of a finite series of numbers $a_1 + \cdots + a_n$. But any notion that extends to an infinite series necessarily involves some sort of limiting process, so it's not really a sum. So, we have choices about how we wish to make such a generalization.