# Nicole Oresme's Proof That the Harmonic Series Diverges

## The Harmonic Series

The series is called the harmonic series. In general, a harmonic series is any series that can be expressed in the form . Whether or not a series diverges (grows towards infinity), or converges (grows towards a finite number) was argued for many years. Notice that when we write out the first few terms of the harmonic series that each term is getting smaller and smaller: . This tells us this series might converge, but does not guarantee that it converges.

The French mathematician Nicole Oresme (ca. 1323-1382) was the first to demonstrate that the harmonic series diverges. Her method is simple and elegant. It is easy to conclude that >= . Regrouping this series we get: . This can easily be rewritten as: . Since the series 1 + 1/2 + 1/2 + 1/2 + ... does not converge, then the harmonic series can not converge either. As you see, this demonstration has the disadvantage of requiring a proof that a finite series of sequential terms of the harmonic series can always be found that sums to more than 1/2.

## Citation

Cite this source as:
McAdams, David E., "Nicole Oresme's Proof That the Harmonic Series Diverges", from LifeIsAStoryProblem.org, 30 June 2007, URL https://lifeisastoryproblem.tripod.com/numbers/oresmeharmonicseries.html

## Other Resources

An infinite series of surprises by C. J. Sangwin.
http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/pus/infsersup.pdf

Harmonic Series Eric W. Weisstein, From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/HarmonicSeries.html