I’ve been meaning to submit this to the Proofs Without Words column ever since I discovered it way back when I was learning calculus. At the time, I wasn’t very impressed by showing that the harmonic series diverged using integral approximations for some reason. I wish I could remember why — it would probably make me a better calculus teacher. This is what I came up with to show the divergence more directly(?):

I’ll leave the interpretation as a puzzle to the reader.

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July 12, 2007 at 10:50 pm |

Wow, that’s really nice. After I figured it out, I was inspired to make another picture.

There’s two other cute proofs that I know of, although neither comes with a picture. The first is:

The above proof is essentially a discrete version of the integral test. In particular, the nth partial sum is roughly the base-two logarithm of n. There’s some sense in which the integral test actually is the right way to prove that the harmonic series diverges — for example, the partial sums of the harmonic series are a bounded distance from ln(n).

Here’s the other proof, which is more similar in spirit to yours:

July 13, 2007 at 4:44 pm |

After I figured out the picture it’s a very sweet proof. I’m teaching

a summer session course on single variable calculus and will use it

to prove the divergence of the harmonic series. The latter proof in

Jim Belk’s comment is also new to me.

July 15, 2007 at 9:47 am |

The picture illustrates the methods by which medieval thinkers such as

Nicole Oresme (ca. 1323 – 1382) and Robert Suiseth (Swineshead) (fl. ca. 1350) summed or verified divergence of what we would now call infinite series. The usual calculus textbook proof, the first one mentioned by Jim Belk, is due to Oresme, although it was lost and rediscovered many times. Some variant of the picture is likely how Oresme came up with the argument.

A nice collection of proofs of the divergence of the harmonic series can be found here:

July 18, 2007 at 7:46 am |

A very nice proof indeed!

July 22, 2007 at 10:11 am |

Could anyone really produce a rigorous proof from this picture?

July 22, 2007 at 10:55 am |

student, sure you can. Just write down both series of areas of rectangles and it jumps out at you.

August 4, 2007 at 4:38 pm |

I love it!

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