2.1 Avoiding Infinite Summations (continued)

Ordinary sums are very well behaved. They are associative, which means that it does not matter how we group them:

(2 + 3 ) + 5 = 2 + (3 + 5),

and they are commutative, which means that it does not matter how we order them:

2 + 3 + 5 = 3 + 5 + 2.

These simple facts do not always hold for infinite sums. If we could group an infinite sum any way we wanted, then we would have that

(2.1.2)
whereas by changing how we group the summands, we obtain

(2.1.3)
It takes a little more effort to see that rearrangements are not always allowed, but the effort is rewarded in the observation that some very strange things are happening here. Consider the alternating harmonic series

Click here to explore the partial sums of this series.

The partial sums are narrowing in on a value near 0.7, (in fact, this series converges to ln 2).
If we rearrange the summands in this series, taking a positive term, then two negative terms, then the next positive term, then the next two negative terms, and continue rearranging in this manner, we get:

then we have not changed any of the summands; we have only changed their order. But the value of the series has changed. It now approaches (ln 2)/2. Rearranging the summands has changed the value.

Click here to explore partial sums of this and other rearrangements of the alternating harmonic series.

Exercises