Albert r meyer massachusets institute of technology creative commons 2010, prof. Posts about book stacking problem written by dragallur. Your quest is to stack them on top of each other but at the same time try to hang them over side of table as much as you can. The books homepage helps you explore earths biggest bookstore without ever leaving the comfort of your couch. Hn to represent the sum of the first n terms of the series.
Lets investigate more about the infinite harmonic series, as shown below. May 12, 2015 so i looked for a selfstudy book, and book of proof turned out to be the book. That sums up what i was thinking, the harmonic series really, really, really wants to converge if were anthropomorphizing numbers now, but it cant quite get there. To balance one book on a table, the center of gravity of the book must be somewhere over the table. Proving that the harmonic series is divergent physics forums. Alternate proofs of this result can be found in most introductory calculus textbooks, which.
Harmonic stacks became widely known in the recreational math community as a result of their appearance in the puzzlemath book of gamow and stern 5 buildingblocks, pp. This is accomplished through a sequence of s trategic replacements of rectangles with others of lesser area. The block stacking problem has been exposed to a wide audience, and the items being stacked are variously referred to as blocks, books, bricks, slabs, cards and coins. An interesting problem is to determine how far an overhang we can achieve by stacking books on a table, accounting for the force of gravity. According the the p series test, must converge only if.
The harmonic series is counterintuitive to students first encountering it, because it is a divergent series though the limit of the nth term as n goes to infinity is zero. When p 1, the pseries is the harmonic series, which diverges. Here are results for the first dozen rows of blocks. Harmonic series mathematics simple english wikipedia. The root test also does not apply in this scenario. This series converges by the alternating series test. Lets start with the proof why you can actually do this in the first place. So, the same can be said of the harmonic series as well.
In mathematics, the harmonic series is the divergent infinite series. Chen, interesting series associated with central binomial coefficients, catalan numbers and harmonic numbers, j. Place identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang paterson et al. It turns out that the maximum overhang possible for books in terms of book lengths is half the th partial sum of the harmonic series this is given explicitly by. The alternating harmonic series sums to ln 2 c laim. Since the harmonic series is known to diverge, we can use it to compare with another series. One example of these is the worm on the rubber band. Harmonic series mathematics simple english wikipedia, the. The original series converges, because it is an alternating series, and the alternating series test applies easily. According the the pseries test, must converge only if.
Recognizing the sum as the harmonic series, we can simplify this formula. The book stacking problem maple programming help maplesoft. An infinite number of mathematicians walk into a bar the first one orders a beer, the second one orders a half a beer, the third orders a quarter of a beer, and this trend continues on for some time. Its tailormade for selfstudy, and as a bonus, its very affordable. We begin by giving the following estimate for the partial sum of a pseries. The absolute value of the terms of this series are monotonic decreasing to 0.
From a pedagogicalpoint of view, the harmonic series providesthe instructor with a wealth of opportunities. Jillian thinks again and comes up with a more imaginative way of stacking the books. The nth term test and the divergent test may not be used to determine whether this series converges, since this is a special case. It explains why the harmonic series diverges using the integral test for series. Hammack literally starts at the beginning, with the basics of notation, sets, logic, etc. The standard proof involves grouping larger and larger numbers of consecutive terms, and showing that each grouping exceeds 12. Leo goldmakher, a shorter proof of the divergence of the harmonic series. Manipulating harmonic series mathematics stack exchange. Line 3 here the known portion of the sequence is replaced by h 2n. The classical harmonic stack of size n is composed of n blocks stacked one on. Dickau, harmonic numbers and the book stacking problem. Summary we demonstrate graphically the result that the alte rnating harmonic series sums to the natural logarithm of two. The book stacking problem main concept if you were to stack n identical books on.
After a while, the bartender gets fed up and hands them 2 beers, shakes his head and says, you mathematicians just dont know your limits. Harmonic series mathematics wikimili, the free encyclopedia. Suppose lower n, squeezed and upper n are sequences with lower n squeezed upper n for each n. In this note, we provide an alternative proof of the convergence of the pseries without using the integral test. If you have been looking at the harmonic series above closely, you may have noticed that some notes that are written to give the same interval have different frequency ratios. However, here is a more elementary proof of the convergence of the alternating harmonic series. From a pedagogical point of view, the harmonic series provides the instructor with a wealth of opportunities. The harmonic series diverges again and again steven j. The maximum amount of overhang, in terms of book lengths, is given by half of the n th partial sum of the harmonic series. In fact, our proof is an extension of the nice result given by cohen and knight 2. Harmonic numbers and the bookstacking problem back to roberts math figures an interesting problem is to determine how far an overhang we can achieve by stacking books on a table, accounting for the force of gravity. In order to find the number of stacked books required to obtain d. This calculus 2 video provides a basic introduction into the harmonic series. Finding big o of the harmonic series stack overflow.
This proof is elegant, but has always struck me as. How far can a stack of books protrude over the edge of a table without the stack falling over. As a known series, only a handful are used as often in comparisons. Of course this isnt a proof because of the approximations. Harmonic numbers and the bookstacking problem the math forum. Find materials for this course in the pages linked along the left. The bernoullis and the harmonic series mathematics. When you use the comparison test or the limit comparison test, you might be able to use the harmonic series to compare in order to establish the divergence of the series in question. Whenever i teach infinite geometric series to high school students ie. Alternate proofs of this result can be found in most introductory calculus textbooks, which the reader may find helpful. Because the harmonic series diverges, the maximal overhang tends to infinity as n \ displaystyle n n. A proof without words that the sum is was shown by matt hudelson a related series can be derived from the taylor series. Line 4 we now replace h 2n with the inductive hypothesis which we have already proven.
In statics, the blockstacking problem is a puzzle concerning the stacking of blocks at the edge. Johnson 1955, also the bookstacking problem, or a number of other similar terms is a puzzle concerning the stacking. Here youll find current best sellers in books, new releases in books, deals in books, kindle ebooks, audible audiobooks, and so much more. Every term of the series after the first is the harmonic mean of the neighboring terms.
This is fascinating problem and i want you to try to take twenty cards or same blocks. Applications such as gabriels wedding cake fleron 1999 and eulers proof of the. This can produce a larger overhang indeed, but unfortunately uses more than one book per level. So, for a stack of n books, the largest possible distance from the edge of the table to the leading edge of the n th book is. A recent proof due to leonard gillman starts with a contrary assumption that the series \sum 1n converges to a finite number s. Proof 24 a limit comparison proof in the last proof the harmonic series was directly compared to the divergent series x.
Stack overflow for teams is a private, secure spot for you and your coworkers to find and share information. Thus the sequence of partial sums of the harmonic series exceeds termbyterm the sequence of partial sums of a series that diverges to infinity. Harmonic numbers and the bookstacking problem robert dickau. Hence, using the definition of convergence of an infinite series, the harmonic series is divergent. By an argument made famous by leibniz the alternatingseries test, we can conclude that the alternating harmonic series converges. The problem of stacking a set of objects, such as bricks, books, or cards. Dickau, harmonic numbers and the bookstacking problem. As you continue you will find that this goes in particular series called harmonic series. A simple and elegant way to stack identical flat objects so that they project. The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the n th term as n goes to infinity is zero. For example, the interval between the seventh and eighth harmonics is a major second, but so are the intervals between 8 and 9, between 9 and 10, and between 10 and 11.
So we see that although the alternating harmonic series converges,the series. Previous authors have thought of blocks as cubes, books, coins, playing cards, etc. Jillian quickly has the idea of simply stacking books vertically near the edge of the. Hn as the total overhang is represented by the sum of 1 2n, not 1 n. Get everyone to agree that the sum isnt more than one. Harmonic numbers and the bookstacking problem an interesting problem is to determine how far an overhang we can achieve by stacking books on a table, accounting for the force of gravity. So i looked for a selfstudy book, and book of proof turned out to be the book.
If you write out the first few terms, the series unfolds as follows. In mathematics, harmonic series sums occur so often, they are given their own symbol. Mar 28, 2018 this calculus 2 video provides a basic introduction into the harmonic series. Because of their association with the intervals found in a harmonic series, intervals such as.
Either the integral test or the cauchy condensation test shows that the pseries converges for all p 1 in which case it is called the overharmonic series and diverges for all p. The block stacking problem is the following puzzle. Can you derive a formula for the maximum overhang possible for n blocks. Maximum overhang dartmouth mathematics dartmouth college. Citeseerx prairie state college the harmonic series. Therefore hn is unbounded, and the harmonic series diverges. It turns out that the maximum overhang possible for books in terms of book lengths is half the th partial sum of the harmonic series. Because the harmonic series diverges, the maximal overhang tends to infinity as n \displaystyle n n. The divergence of the harmonic series is also the source of some apparent paradoxes. The harmonic series is the sum from n 1 to infinity with terms 1 n. Mathematicians will recognize this as a harmonic series. Hall gives a sequence of stacks which he claims, without proof, to be. A path to understanding musical intervals, scales, tuning and timbre. Harmonic numbers and the book stacking problem back to roberts math figures an interesting problem is to determine how far an overhang we can achieve by stacking books on a table, accounting for the force of gravity.
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