arrow_back Here's an example. You know that sometimes it's OK to add up infinitely many things (though you can't literally do just that: you don't have infinite amounts of time to spend---so you do need limits here). So for example you can add up 1+1/2+1/4+1/8+1/16.... to get 2. On the other hand, its pretty clear that if you were to add up 1+2+3+4+.... then you wouldn't get any real number. You might say that what you got was bigger than any number, and so was infinity. But you'd be pretty clear that it wasn't any ordinary number.

What about 1+(-1)+1+(-1)+1+(-1)+....?

Let's suppose you could add that up to get s. What is s?

Well, you could write

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

But you could also write

s=1+( (-1)+1) + ((-1)+1) +.....)=1.

Or you could figure

s-1=(-1)+1+(-1)+1+(-1)+....=-s

so that s=1/2.

In fact, there's a good case to be made for s being any real number at all! (We'll see this is more detail when we come to study series.)

Of course, what all this tells us is that it's absurd to think that this sum adds up to anything. But that means that some infinite sums can be "added up" and others can't. So, again, we need to have a clear idea of what goes into making infinite sums work---a clear idea of limits.