Look at it like this. (-x)^2 is positive. (-x)^3 is negative. In general, (-x)^2n is positive whilst (-x)^2n+1 is negative. The reason it is undefined is because of the fact that is infinity even (therefore (-x)^infinity is positive) or odd (negative)?Why is (negative constant)^infinity undefined?
*Sketches graph on Geogebra and investigates*Actually, any number (positive or negative) to the power of infinity is undefined.
Even 1 to the power of infinity is undefined (people think it's still 1).
Tell me what you think about this 'proof':
my head almost exploded when my lecturer showed me thisI remember seanieg89 saying something really funny before... something along the lines of:
"What is 1 to the power of chair?"
to prove his point.
Essentially, infinity isn't a number and can't be treated as such. Anything dealing with infinitismals is VERY sensitive.
ie: Consider an infinite series. If it is 'Conditionally Convergent', I can actually make it converge to ANY value I want it to with a specific permutation of the terms. But it is a permutation of INFINITE terms. This is called Riemann's Arrangement Theorem.
You can even make it diverge! A classic example of this is that by manipulating the Alternating Harmonic Series (which converges to ln2), we can make it converge to say 3/2 ln(2), which is most certainly false.