 Finite sums can be regrouped and rearranged. 1 plus 2 plus 3 plus 4 is the same as 1 plus 4 plus 2 plus 3. Can we do that with infinite series? Well, let's try it. So I suppose we start off with the series of reciprocals, and we note that 1 over 2n is, and 2 times 1 over 2n is. So let's see if we can rearrange them. What if we take this series and subtract 2 times 1 over 2n? Then if we regroup them, we have 1, then we have 1 half minus 2 times a half, the 1 third, 1 quarter minus 2 times a quarter, and so on. And we can simplify. Now the series on the right turns out to be equal to log 2. And 2 times 1 over 2n is the same as 1 over n. And so our right-hand side is log 2, and our left-hand side is 0. And if we exponent... That was a general rule. 2 is not equal to 1, so we might wonder what went wrong. And in this case, note that the series we started with is a divergent series. Consequently, we have the rule, never work with divergent series. Well, what if we work with a convergent series, like how about the series for log 2? Now let's rearrange the terms a little bit. And here we note that all of our odd-denominator terms are going to be added, and our even-denominator terms are either half of an odd-denominator term, or a multiple of 4, like 1 fourth, 1 eighth, and so on. And so if we rearrange our terms this way, we get... And if we do a little bit of arithmetic, we find... And let's rearrange these terms so the denominators are increasing. And everything here has a factor of 1 half, so we'll factor that out. And here's our familiar series for log 2 again. And so we find that log 2 is 1 half log 2. Yeah, we don't like this either. And so we see there's a problem. Rearranging the terms of a divergent series or a conditionally convergent series leads to a contradiction. The only type of series left is an absolutely convergent series. And so the question to ask is, what happens if we rearrange the terms of an absolutely convergent series? The answer is... And we'll see why in the next lecture.