The proof that 1/2+1/4+1/8+...=1 goes like this: X=1/2+1/4+1/8+... 2X=2/2+2/4+1/8+... 2X=1+1/2+1/4+... 2X-X=1+(1/2-1/2)+(1/4-1/4)+... X=1 The assumption that goes with this is that we can pair up the first term of X with the second term of 2X and so on without having the smallest term of X leftover. This is because there are infinitely many terms and ∞+1=∞. The result seems intuitively satisfying, the term that is "ignored" is after all infinitely small. But lets look at another example where we "ignore" the largest term: X=1+2+4+... 2X=2+4+8+... 2X-X=-1+(2-2)+(4-4)+... X=-1 So here we see that we get a negative number from adding up only positive numbers. Also we get a finite number from something that goes towards infinity. This is intuitively not very satisfying in my opinion and ought to be a proof that ∞+1=∞ doesn't work. After all X is also an infinity ∞, so: ∞+1=∞ -1+1=-1 0=-1 Also since: ∞=∞+1 We can add as many as we like ∞=∞+1+1+1+... ∞=∞+∞ Again we can keep adding ∞=∞+∞+∞+∞+.... ∞=∞*∞ ∞/∞=∞ This is not only not intuitive, it is completely useless, since anything "tainted" by this type of infinite, will be impossible to calculate anything with. Also lets try the pairing up method for 1...000 and 0 1...000=...000=0 Every zero in the large number can be paired up with every zero in 000... So we have both ∞=0 and ∞=-1 again 0=-1 If on the other hand we say: ∞≠∞+1 We get an infinite amount of infinities that are not equal. Further if we write the infinite in this way: ∞=X+Y+Z+... ∞/∞=1 because there will always be an equal amount of terms (infinite or not). That also means that X=2∞ X/∞=2 So one infinite divided by another can be any finite. Also ∞*∞=X X/∞=∞ That means that dividing infinities can also get you an infinite. Lets try to assume that an infinite divided by a finite X could yield another finite Y: ∞/X=Y ∞=YX Since the product of two finite numbers can never be infinite, we know this is incorrect. Also we can also say something about the infinities divisibility, like 2+2+2+... is divisible by 2. Anyway now we get different results for a lot of calculations: 1/2+1/4+1/8+...=1+1/∞ (where ∞ is the number of terms) X=1+2+4+... = 2^(∞-1)-1 1=0.999...+0.1^∞ (∞ being the number of nines) If you have an infinite hotel fully booked, you cannot fit any more people in. This makes much more sense to me. What do you think?