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Sunday, June 12, 2005

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.


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