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I Like Abstract Stuff; Why Should I Care?

You've known for a long time that addition is commutative. No matter what order you add the numbers, you'll always get the same answer. Well... that's true when you're adding finitely many numbers.

When you're adding infinitely many numbers, the rules change. If a series converges absolutely you can still rearrange the terms however you want. But if a series converges conditionally, rearranging the terms can make them add up to different values. In fact, the terms can be rearranged to add up to anything you want.

The alternating harmonic series is a good example of this weirdness. The alternating harmonic series is conditionally convergent, and when we get to Taylor series we'll see that it sums to ln 2. In symbols,

Now let's rearrange the terms. We can write the terms of the alternating harmonic series like this:

First take a moment to convince yourself that all the terms of the alternating harmonic series do show up in this rearrangement.

Then simplify the stuff in parentheses to get

Factor out  from each term (that is, factor out 2 from each denominator). Now we have

This looks familiar. It's  multiplied by the alternating harmonic series. Since the sum of the alternating harmonic series is ln 2,

By rearranging the terms, we made them add up to only half of what they added up to before.

That was only one example. What if we wanted to make the terms add up to 7? Or -32? This article gives a nice explanation of why we can make the terms add up to anything we want:

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