Originally Posted by
Eviscero
Holy wow this is such a silly thing to do. Step 1 and they have already made a very strange choice. They define the infinite sum
S1 = 1-1+1-1+1 ... to be 1/2 because it is the AVERAGE of the two partial sums that can possibly be achieved: 0 and 1. Okay, fine... you're welcome to do that. I much prefer the standard approach which is to say an infinite sum is said to converge if the limit of the sequence of partial sums converges (that is define the sequence s(k) = the sum of the first k terms. If for every number epsilon > 0 there exists a natural number N such that |s(N) - L| < epsilon, we say the infinite sum "is equal" -- really converges -- to the number L). If no such L exists we say the sequence diverges (in other words is not equal to anything). There are some very interesting results about when an infinite sum converges and it should be pretty clear that S1 does not converge according to the definition proposed above. There are many reasons why the above definition "is the right one" and I am very curious to know in what context their averaging definition "is the right one."