Rearrangements of a Conditionally Convergent Series Summing to Logarithms of Natural Numbers

Click to increase image sizeClick to decrease image size

Acknowledgment

The author is grateful to the editorial board for their helpful suggestions. In particular, the board pointed out that some partial sums are, in fact, Riemann sums, which is the approach used in this note. The author is also grateful to his colleague, Phuc Cong Nguyen, for reading a preliminary version of this note.

Summary

This article gives a nice family of rearrangements of a conditionally convergent series for use in calculus or first analysis classes. In each series, the positive terms and the negative terms both form a harmonic series. For each natural number k, a series is given that converges to the logarithm of k. The series are wonderful to show to any calculus class even if the instructor omits the details. The series or variations are known.