Boas, R. Monthly 78 , , Borwein, J. Experimentation in Mathematics: Computational Paths to Discovery. Derbyshire, J. New York: Penguin, pp. DeTemple, D. Gardner, M. Hardy, G. New York: Chelsea, Havil, J. Hoffman, P. New York: Hyperion, p.
Honsberger, R. Washington, DC: Math. Rosenbaum, B. Monthly 41 , 48, Shutler, P. Sloane, N. Wells, D. Middlesex, England: Penguin Books, p. Weisstein, Eric W. If it breaks, record its breaking strain. If it survives, go on the next beam. This procedure successively finds the record minimum breaking strains. Only beams with record low breaking strains are destroyed by the test.
Out of one hundred beams, you expect to destroy 5. If you had a thousand beams to test, you would expect to destroy only 7. Just about the simplest mathematically speaking way of shuffling cards is called the "Top in at random" shuffle. The top card of a card deck of cards is removed and inserted at random in the deck. How many times must this shuffle be repeated before we can regard the deck as "random"?
Let us follow the progress of the card which is initially at the bottom of the deck. This card label it stays at the bottom until another card is inserted below it. Since there are places into which a card taken from the top can go, the chance that it will go below is , and therefore on average it will take "top in at random" shuffles before a card is placed below.
Now the chance that a card taken from the top and inserted at random into the deck will go in below is since there are now two places below , and the expected number of shuffles needed to get a second card below is.
Thus the expected number of shuffles needed to get two cards below is. Note that at this stage the cards below are in random order. Continuing in this way, we see that the expected number of "top in at random" shuffles needed to get up to the top of the deck is At this stage the cards below are in random order, and just one more shuffle, which puts at random into the deck, is needed to randomise the deck.
The total number of shuffles needed is thus. You have to cross the desert in a jeep. There are no sources of fuel in the desert, and you cannot carry enough fuel in the jeep in order to make the crossing in one go. You haven't the time to establish fuel dumps, but you do have a large supply of jeeps. How can you get across the desert, using the minimum amount of fuel? Let us measure the distance a jeep can travel in terms of a tankful of fuel.
One jeep by itself can travel a distance of one tankful. Between them, they have enough fuel to get back to base. Here we have a new series, which is also harmonic the reciprocals are in arithmetic progression , and also diverges, for clearly The fact that the series diverges shows that, by using the system of transferring fuel, you can effect a crossing of arbitrary length, as long as you have enough jeeps.
One might add: as long as the desert is wide enough to accommodate all those jeeps! We have just seen that the series which is obtained from the original harmonic series by deleting every second term, still diverges.
What about the series where the first term and all terms with composite non-prime denominators have been deleted? Since the remaining denominators are all prime, and the prime numbers are very thinly scattered, it is indeed surprising that the series of reciprocals of primes still diverges.
The proof of this fact is a little too complicated to give here although it's only first-year university level so I'll leave you to try to discover it yourself. When you've proved that this series diverges, you can deduce as a corollary that there are infinitely many prime numbers, though there is a simpler way of proving this fact.
Instead of deleting all terms with composite denominators, let's delete every term which has a zero in its denominator. It looks as though one is deleting roughly one in ten of the terms of the original harmonic series. Thus it seems a reasonable guess that the leftover series diverges, and it may be a shock to your intuition to learn that your guess is wrong. Let us look first at all terms of the leftover series with just one digit in the denominator. There are exactly 9 of these, and they are all less than 1.
Their sum is thus less than 9. Next, look at the terms of this series with exactly two digits in their denominator. Their sum is less than. In general, there are terms of the series with exactly digits in the denominator, and each is less than. Their total is thus less than. Thus the sum of the terms in the series is less than which is a geometric series whose sum to infinity is Thus the harmonic series without the terms containing zero digits converges.
A more careful analysis can be given to show that the sum of this series is Let us now go back to Oresme's proof that the harmonic series diverges, which was achieved by showing that. What Pythagoras was experiencing was what we now call the harmonic series.
Through experimentation with the blacksmith and his three assistants, Pythagoras was able to draw out a series of ratios between physical dimensions and pitch, which we still use today to describe how specific sounds are related on our instruments. Above are all the notes that are playable on the open F horn.
Valves help fill in the notes in between to allow a fully chromatic scale. Chromaticism is an artificial construct within the natural overtone series. When you use a different valve combination, the harmonic series has the same intervallic relationships because the interval sequence is always the same.
For instance, the interval from the 2nd to 3rd harmonic is always a fifth. As you ascend, the intervals become narrower. As brass players, you might recognize some of your favorite lip slurs as a part of the harmonic series above.
Lip slurs are a basic fundamental you likely began with when you started your horn. You can slur through the partials in order, or you can skip through them when playing larger intervals. Some of the partials on the harmonic series are naturally out of tune.
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