[split] 0.999... equals 1

[Tiberius]

Any number raised to the power of infinity is undefined.

True.

0<x<1 raised to the power of infinity will converge and the limit is zero, but the number is not zero.

False. As previously stated, due to the completeness property of all real numbers, each "limit" is itself a real number. The limit of the geometric sequence 0.9 + 0.92 + 0.93 + ... is 1, and hence the sequence (representing the number 0.999...) is also equal to 1.

See: http://en.wikipedia.org/wiki/Real_number#Completeness

Infinity + 1 or +2 or + any number still equals infinity.

No it doesn't. Infinity is not a number that can be used in arithmetic calculations.

Any formula based on infinity deals with undefined terms, only the limit is valid, but the result is just that, a limit.

Unless that limit is dealing with a geometric sequence determined from a real number, in which case the limit is also a real number.

The proof for the geometric series is a limit based on this undefined number, so the geometric series formula is technically invalid. And again, it is acceptable at this time until we come up with a better way to handle infinities.

No. You can hold that the geometric series formula is "invalid" if you want, but mathematicians do not. They hold it as perfectly valid, and there are various proofs of its validity. It is not just something we "accept" until we come up with a better way to handle things. Mathematics is not science; it doesn't deal with empirical methods, but rather ones of logic. How mathematics works is how our logic determines it to work, and how we've designed it to work. The quirks we get out of mathematics aren't going to change.

Infinity+1=infinity subtract infinity from both sides 1=0 This works with any equation in one form or another, ergo, undefined.

Again, you cannot use infinity in arithmetic calculations; it is undefined and so your calculation above is wrong.