# Graphical Representations of Depleted Zeta Subseries

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It is well-known that the harmonic series diverges; equivalently, equals infinity. Remarkably, modified forms of , denoted here by , can yield various converging values. For example, deleting all terms in the harmonic series whose denominator contains a 9 converges to approximately 23; this is known as the Kempner sum. More generally, represents a subseries of with terms deleted whose denominator's base representation contains the string of digits given by .

Contributed by: Andrew Kwon (December 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The initialization code is copied from code (based on [1]) by Thomas Schmelzer and Robert Baillie: http://library.wolfram.com/infocenter/MathSource/7166/#downloads. Reference

[1] T. Schmelzer and R. Baillie, *American Mathematical Monthly* 115(6), June/July 2008, pp. 525-540.

## Permanent Citation

"Graphical Representations of Depleted Zeta Subseries"

http://demonstrations.wolfram.com/GraphicalRepresentationsOfDepletedZetaSubseries/

Wolfram Demonstrations Project

Published: December 2 2015