Infinite Series Part III

So, we see that in order for an infinite sum to make sense, we must use limits.

So, that is how an infinite sum is defined: if Sn is the "n'th partial sum" (what you get when you sum the first "n" terms with ordinary addition, then the infinite sum is the limit of Sn as n goes to infinity, provided the limit exists.

So that is why we need limits and sequences to talk about infinite sums.

Now what about that strange infinite sum 1 - 1/2 + 1/3 - 1/4 + 1/5......

This is an example of an alternating sum because the signs alternate. To find what this sums to, one must study infinite series. But to see that this sum exists:

Let S2n+1 be the odd partial sums. Since each S2n+1 is a finite sum, we can group and sum the terms in any order; we get (1-1/2) + (1/3-1/4).....+ 1/(2n+1) =

1/2 + 1/12 + .....(1/(2n-1) - 1/2n) + 1/(2n+1) =

1/2 + 1/12 + ....((2n - (2n-1))/(2n(2n-1)) + 1/(2n+1) =

1/2 + 1/12 + 1/(4n^2 -2n) + 1/(2n+1)

We need some results about infinite series here, but as n goes to infinity in we get a convergent

series (p test and comparison test). That is why we have all of those tests.

Now here is what is bizarre: if we change the order of the original series, we change the result of the sum! This is not obvious, and to see this we need some infinite series results.