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.

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,

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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