Summation Woes

Well, here's something that both JX and Luoning can understand and appreciate; it's basically something I discussed with Shaun over MSN, and it's an alternating series, taking the form of:


Notice that we can group the terms like this and obtain a sum for an odd number of terms:


Also, we can group the terms like this and obtain a sum for an even number of terms:


The strange thing is, notice that if you consider an even number of terms, the sum goes to positive infinity, and if you consider an odd number of terms, the sum goes to negative infinity! This is what we call an oscillating series, which happens to be diverging as well.

I’ve included a graph for your reference, to illustrate the oscillating and diverging nature of this series:


Notice the diverging nature of the sum to n terms, and consequently, there can be no sum to infinity, simply because the sum to n terms depends on the very number of terms, and thus on the last term. A sum to infinity where the number of terms and the last term is not defined can therefore produce no well defined result.