Harmonic Digression


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.

8 Responses to “Harmonic Digression”

  1. Jim Belk Says:

    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:

    \displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \cdots

    >\displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{16} + \cdots

    = \displaystyle\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots = \infty

    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:

    \displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \cdots

    >\displaystyle \frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{6} + \frac{1}{6} + \frac{1}{8} + \frac{1}{8} + \cdots

    = \displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

  2. ulfarsson Says:

    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.

  3. Michael Kinyon Says:

    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:

    Click to access harmapa.pdf

  4. Grétar Amazeen Says:

    A very nice proof indeed!

  5. student Says:

    Could anyone really produce a rigorous proof from this picture?

  6. John Armstrong Says:

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

  7. Donna Dietz Says:

    I love it!

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