A mathematical series known to diverge by the integral test. It comes in the form 1+1/2+1/3+1/4+1/n. This can be used to compare another series, if every term is greater than the harmonic series, that series will diverge.

Another way of showing the harmonic series diverges, without all those fancy comparisons to integrals (and having to know that limx->infinity log(x) = infinity, which is somewhat harder to prove), is to perform the following elementary comparison.


1 >= 1
1/2 >= 1/2
1/3 >= 1/4
1/4 >= 1/4
1/5 >= 1/8
1/6 >= 1/8
1/7 >= 1/8
1/8 >= 1/8
...


Generally, each of the terms 1/(2k+1), 1/(2k+2), ...,
1/2k+1 is at least 1/2k+1, and there are 2k such terms. Thus, the sum of these terms is at least 1/2, so the sum of the first 2k terms of the harmonic series is at least k/2; thus, the series diverges.



Well actually, I lied. This sort of thing is called the condensation test for series convergence...

With reference to sound, the harmonic series is a series of frequencies which can be derived back to a fundamental frequency. For example you could have a piano string vibrating at 256Hz or you could double that and the string's vibration would have a frequency of 512Hz. Both situations are possible on that string:

256Hz:
    _
/ \
/ \
\ /
\ _ /
512Hz: _ _
/ \ / \
/ \ / \
\ / \ /
\_/ \_/
This occurs mainly in standing wave setups such as brass instruments, some wind instruments, the piano (to a small degree), etc.

Tonality of music is based upon the harmonic series because when you hear a note the hairs in your ear vibrate at that frequency whilst some hairs vibrate at a frequency in the harmonic series of that note. That is why some chords make you cringe and others sound nice - because the former is messing around your sensitive hairs in your ear and the latter is making them all vibrate together harmoniously.

For a good example of harmonics in action go to a real piano and press down and hold a low C slowly so as to not make any noise. Then play a C one octave higher quickly. Notice how you can stil hear it playing even after you have let go of that top c?
That is because the bottom C is now vibrating at the top C's frequency.

Convergence Of The Rearranged Alternating Harmonic Series

Although the harmonic series doesn't converge, the alternating harmonic series
1 - (1/2) + (1/3) - (1/4) - ...
does, and it converges to the natural logarithm of 2.

This fact can be generalized in an interesting way. If we rearrange the harmonic series so that we take, alternately, r positive terms and s negative ones, the series converges to (1/2) ln (4r/s).

The proof of this, for me, took roughly four pages, but the basic gist of it is this:

That part is simple enough. It just takes some slightly clever algebraic manipulation and the knowledge of the Taylor series for ln (1 + x), which is
ln (1 + x) = x - (x2)/2 + (x3)/3 - . . .
The hard part comes with filling in the details. The hardest part is showing that the series is continuous at x=1, so that we can take the limit in the last step and know that it's the same as the actual value of the series at 1. In order to show that a series of functions is continuous, we need to show that it uniformly converges there, so this is not trivial.

This result is slightly less interesting in light of the fact that any conditionally convergent series can be rearranged to converge to any real number, but it is kind of neat nonetheless.


Note: I did a project on this for my Real Analysis class, but this node is not cut and paste. Thanks to David Bressoud of Macalester College for assigning it.

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