Here's a neat example of a reasoning taken to be a paradox in 18th century (in relation to infinitesimal calculus). 1. Consider the infinite sum: 1/2 + 1/4 + 1/8 + ... + 1/2 n + 1/2 n+1 ... Intuitively, it won't go above 1, and will be higher than any number below 1. So, we might take it to be 1. 2. Now consider: 1-1+1-1+1-1+... One way to think about this sequence is this: (1-1)+(1-1)+(1-1)+... This way, it becomes 0+0+0+..., and hence it should equal 0. 3. Another way to break it down is to observe that -1+1=-(1-1) and substitute accordingly: 1-(1-1)-(1-1)-(1-1)... But if we count this way, we get 1-0-0-0-0.... = 1. 4. Now, call the whole sum U. U= 1-1+1-1+1-1+... The reverse: -U will be obtained by reversing the signs before the summands: -U= -1+1-1+1-1+1... Add now 1 to each side: 1-U= 1-1+1-1+1-1+1... But this is nothing but 1-U=U! Thus, it should follow that U=1/2. The question some 18th-century mathematicians asked: which one is it?