What is an infinite series?

This entry is designed to help a student understand what an infinite series is and why we need to be fussy.

• Regular Addition

Not a problem; we "know" what, say, "3 + 4" means, or even "4 + 7 + 5" means, etc. In a similar way, we have an intuitive idea of what "x + 2x^2" means. A sum with any finite number of terms is always well defined.

• Infinite Addition: where the problem begins.

So, now what does it mean to add an infinite number of terms together? Let's look at the following examples:

1. 1 + 2 + 3 + ....+ n + (n + 1) +....
2. 1 + 1/2 + 1/3 +.....+ 1/n + 1/(n+1) +......
3. 1 - 1 + 1 - 1 + 1 - 1....... + (-1)^(n)+ (-1)^(n+1) +....
4. 1 + 1/2 + 1/4 + 1/8 + .....+ 1/2^(n) + 1/2^(n+1) + .....
5. 1 - 1/2 + 1/3 - 1/4 + 1/5 +.....+ (-1)^n)*(1/(n+1))+ (-1)^(n+1)*(1/(n+2))

The first thing to notice is that an infinite sum need not make sense at all! First note that it should be clear that the first sum not make sense. After all, it is clear that as one is adding an infinite number of terms which are getting infinitely large.

To see this we introduce the concept of the partial sum; the first partial sum is 1, the second is 1+2 = 3, the third is 1+2+3 = 6. Exercise: what is the fourth partial sum?

So, we look at the partial sums as a sequence: 1, 3, 6, 10, 15, 21....

and we see that the partial sums are getting bigger without bound. Clearly this infinite sum can't make sense.

Now look at example three: the first partial sum is 1, the second is zero, the third is 1, the fourth is zero, and so on. Exercise: what is the (2k+1)'st partial sum? The 2k'th?

Here the sequence of partial sums (1, 0, 1, 0, 1, 0...) are never going to "steady out" on anything. So this sum can't make sense either.

Ok, what about examples 2, 4, and 5? That is the topic of the next blog entry.