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.

Define

By the alternating series test, this series converges. However,

,

which is the divergent harmonic series which was discussed earlier in this series. Therefore, the series converges conditionally and not absolutely.

To calculate the value of , let , the th partial sum of . Since the series converges, we know that converges. Furthermore, the limit of any subsequence, like , must also converge to .

If is even, so that and is an integer, then

.

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

,

or

.

Since and , we have

.

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.

In yesterday’s post, I showed that if

,

the Euler-Mascheroni constant. Therefore, the limit of any subsequence must converge to the same limit; in particular,

.

Applying these above, we conclude

,

or

.

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