The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as
$$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$
In some situations, we can find the result of a divergent series, which should agree with other summation methods.
For example,
$$(1+1)^{-1}=1-1+1-1+1-1+\cdots$$
The right side does not converge, but the left side does, and it agrees with other results.
Another common one is
$$(1+1)^{-2}=1-2+3-4+5-6+\cdots$$
Again, we find a result that agrees with other results.
I was wondering if there were any other interesting divergent series that were rewritable into a binomial expansion and what they were. Maybe something like $(1+1+1)^{-1}$? I don't quite know how to do that.