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. 13231382) 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.

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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 MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicSeries.html
