Question

I'm looking at Grandi's series and trying to figure out what you can't get it to converge to. Grandi's series is:

1-1+1-1+1-... to infinity

Answer 1

So for example you can look at it as:

S=(1-1)+(1-1)+...=0
S=1+(-1+1)+(-1+1)+...=1

or something like:

S=1-1+1-1+...
1-S=1-1+1-1+...
1-S=S
1=2S
S=1/2

But you can do more like:

S=(1+1)-(1+1)+(1+1)-...=2

or

S=S^2=S^3=S^4...

I'm trying to see if I can't make it equal pi, e, or i. Any thoughts or ideas or am I alone on this one haha.

Answer 2

you could make it equal to \(\pi\) or \(e\) but not \(i\) (I think...); playing with divergent real series (since they have no actual sum) can yield any real number, just like how in logic you can assume inconsistent axioms and derive anything: http://en.wikipedia.org/wiki/Principle_of_explosion