In recent posts, we’ve seen the curious phenomenon that the commutative and associative laws do not apply to a conditionally convergent series or infinite product. Here’s another classic example of this fact that’s attributed to Cauchy.

We’ve already seen in this series (pardon the pun) that

.

Let’s now see what happens if I rearrange the terms of this conditionally convergent series. Let

,

where two positive numbers alternate with a single negative term. By all rights, this shouldn’t affect anything… right?

Let be the th partial sum of this series, so that contains positive terms with odd denominators and negative terms with even denominators:

.

Let me now add and subtract the “missing” even terms in the first sum:

.

For reasons that will become apparent, I’ll now rewrite this as

,

or

Since , , and , we have

,

or

.

I now take the limit as :

.

This step reveals why I added and subtracted the integrals above: those gymnastics were necessary in order to reach a limit that converges.

As shown earlier in this series, if

,

the Euler-Mascheroni constant. Therefore, since the limit of any subsequence must converge to the same limit, we have

Applying these above, we conclude

,

which is different than .

Technically, I’ve only shown so far that the limit of partial sums 3, 6, 9, … is . For the other partial sums, I note that

and

.

Therefore, I can safely conclude that

,

which is different than the original sum .

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