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Harmonic sum and a stack of blocks (check my work)

  1. Apr 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Let S_n = 1 + (1/2) + (1/3) + (1/4) + ... + (1/n)

    a) What value of n is required such that S_n > 100?

    b) Show how one can form a stack of identical blocks (one per row), each 1 ft long, so as to form a "wing" (I don't know what the correct word in English is, but I believe you understand) that is 50 ft long. Is it possible to form a "wing" 50 million ft long?

    Taken from Algebra and Trigonometry with Analytic Geometry, by Varberg and Fleming (Spanish version.)

    2. Relevant equations

    In calculus, it is proved that

    ln n < 1 + (1/2) + (1/3) + (1/4) + ... + (1/n) < 1+ ln n

    3. The attempt at a solution

    a) The first thing that I have done is plug ln n = 100. Solving for n one gets it equals e^100. But n must be an integer, so I have two options.

    Let f(x) denote the greatest integer less than or equal to x, and g(x) denote the smallest integer greater than or equal to x.

    If I plug n = f(e^100), I wouldn't know for sure whether S_n > 100. On the other hand, it is certain that n = g(e^100) satisfies the condition.

    b) If there is an integer n such that S_n - 1 = 50, we're done: we can put the second block in the stack so that it sticks out 1/n ft in relation to the first, the third block sticking out 1/(n-1) ft in relation to the second, and so on until we put the top block, which will stick out 1/2 ft in relation to the one that's below.

    If such an integer n doesn't exist, we plug n = g(e^51) (remember the function I defined in part (a):)

    (ln g(e^51)) - 1 < S_n - 1 < ln g(e^51)

    so we can multiply the expression at the center by a number that is less than one in order to get that expression to be equal to 50; that is, we multiply 1/2 , 1/3 , 1/4 , ... , 1/n by that number so as to keep the stack of blocks "harmonic".

    The answer to the second question in part (b) is yes, we can. The procedure is similar to the one described for the 50 ft "wing." The number n in this case will be much bigger.
     
  2. jcsd
  3. Apr 29, 2012 #2
    Hi. Gauss invented function expressing maximum integer that does not exceed x. It may help you.
     
  4. Apr 29, 2012 #3
    Hi, sweet springs. Thank you for replying. Isn't f(x), which I have defined above (of course, I'm not the first person to have used such function), the function you are talking about?

    If I plug n = f(e^100), S_n is not going to be necessarily greater than 100. We don't know what value it is (unless we compute it), but it may be below 100 given that ln x is an increasing function (e^100 is greater than or equal to f(e^100))
     
  5. Apr 29, 2012 #4
  6. Apr 29, 2012 #5
    Honestly, I didn't know their names. On the other hand, I have searched for a solution to the problem on other resources. I see that I have to take into account the center of mass, a concept I am not familiar with (I'm ignorant of physics).
     
  7. Apr 29, 2012 #6
    One of the resources (Wolfram MathWorld) states that the maximum length that can be achieved by n blocks is (1/2)*S_n.
     
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