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Math Help - Problem 48

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    Problem 48

    1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
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    Quote Originally Posted by ThePerfectHacker View Post
    1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
    Let k ΞZ such that 2^k £ n < 2^k+1

    Let m be the least common multiple of 1,2,3,…,n except 2^k.

    Then multiplying S = 1 – 1/2 + 1/3 -…..± 1/n by m we have:

    mS = m – m/2 + m/3 -……± m/n

    Each number on the right hand side is an integer except m/2^k and hence Sm is not an integer, which implies Sm is not an integer.
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  3. #3
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    By the alternating series theorem, the partial sum will always be less than one but greater than zero, and therefore not an integer.
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    Super Member Aryth's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
    The series you presented is the Alternating Harmonic Series, which is Conditionally Convergent, the series is represented by:

    \sum_{n=1}^{\infty} \left(\frac{(-1)^{n+1}}{n}\right)

    The series' terms look like such:

    1 - \frac{1}{2} + \frac{1}{3} - ... \pm \frac{1}{n}

    This series converges to \ln{2}

    Since the series converges to \ln{2} and since:

    |a_{n+1}| < |a_n|

    Then for n \geq 2 the series can never reach one since it is incrementing up or down by smaller amounts. Since you subtract \frac{1}{2} from 1 for n=2, and since the terms are decreasing and alternating in sign, then the series will never reach one again, therefore, this can't be an integer for n \geq 2 because all terms are decreasing,therefore the partial sums remain between 1 and 0.
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    Member Danshader's Avatar
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    how about considering:
    1 +1/2 + 1/3 + 1/4 + .... ===>A
    and
    1/2 + 1/4 + 1/6 +.... =====>B

    to get the required series:
    A - 2B
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  6. #6
    Super Member Aryth's Avatar
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    Yeah, that is a distinct possibility...

    The Alternating Series does equal:

    H(n) - H(2n)

    Where H(n) is the n-th harmonic number
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  7. #7
    Member Henderson's Avatar
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    Huh. I stayed away from this one because I didn't pick up on the series alternating- I read \pm \frac{1}{n} as saying each term could either be added or subtracted, without nessecarily alternating.

    Is there a similar solution to this problem?
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  8. #8
    Super Member Aryth's Avatar
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    You can't know what sign the last number of the series is going to be, that all depends on n, so the \pm means that it can be positive or negative depending on n. The initial pattern reveals an alternating series.
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    [FONT='Cambria Math','serif']My first thought was to try an inductive argument, but I had a lot of difficulty getting it going. I don’t think what I came up with is sound, but nevertheless I decided to post what I came up with.

    Proof. It suffices to show that for all
    n≥2; 1-1/2+1/3-…±1/n ∈ (0,1).
    Let Pn denote the proposition that
    1-1/2+1/3-…±1/(n-1) ∈ (0,1)
    and
    1-1/2+1/3-…±1/(n-1)±1/n∈ (0,1).
    Then P3 is true since
    1-1/2=1/2∈ (0,1)
    and
    1-1/2+1/3=5/6∈ (0,1)
    Assume Pn is true and that n is even. Then
    1-1/2+1/3-…+1/(n-1) ∈ (0,1)
    and
    1-1/2+1/3-…+1/(n-1)-1/n ∈ (0,1).
    Because 1/(n+1) < 1/n, it follows from the inductive hypothesis that
    1-1/2+1/3-…+1/(n-1)-1/n+1/(n+1) ∈ (0,1).
    The case where n is odd is similar. So by the principle of mathematical induction, for all n ≥ 3, Pn is true and hence for all n ≥ 2, 1-1/2 +1/3 -…±1/n ∈ (0,1) and hence not an integer. //

    [/FONT]
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    Quote Originally Posted by Danshader View Post
    how about considering:
    1 +1/2 + 1/3 + 1/4 + .... ===>A
    and
    1/2 + 1/4 + 1/6 +.... =====>B

    to get the required series:
    A - 2B
    These two series do not converge.
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  11. #11
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    Ok, how about this. It is basically Aryth's post but a bit more explicit.

    Let A_N = 1 - \frac{1}{2} + \ldots  \pm\frac{1}{N} which is just the partial sums.

    Consider the (sub) sequence of partial sums:

    <br />
O_N = 1 - \frac{1}{2} + \ldots + \frac{1}{2N+1}<br />
for N\geq1

    O_N is a subsequence of A_N which as noted above converges to ln(2) (derive using MacLauren expansion of ln at x=1). Then O_N\rightarrow\ln(2).

    O_N is monotonically decreasing:

    O_{N+1}-O_N = - \frac{1}{2N+2} + \frac{1}{2N+3} < 0

    Note that O_0=1 and so 1>O_N\geq\ln(2)\approx0.693 for  N>0 and so cannot be an integer.

    Likewise for the partial sums:

    <br />
E_N = 1 + \ldots - \frac{1}{2N}<br />
for N\geq1

    except that E_N monotonically increases from 1/2 to ln(2).

    Put it together and we just showed the odd and even elements of the partial sums A_N are never integers after 1.
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  12. #12
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    Quote Originally Posted by Danshader View Post
    how about considering:
    1 +1/2 + 1/3 + 1/4 + .... ===>A
    and
    1/2 + 1/4 + 1/6 +.... =====>B

    to get the required series:
    A - 2B
    A - 2B = 0? Considering B = A/2...
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  13. #13
    Senior Member bkarpuz's Avatar
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    Exclamation

    Quote Originally Posted by Lore View Post
    A - 2B = 0? Considering B = A/2...
    As meymathis said, these two series do not converge, in other words, A=\infty and B=\infty.
    So you do algebric operations on infinite numbers, which may confuse your mind.
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  14. #14
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    Quote Originally Posted by ThePerfectHacker View Post
    1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
    Suppose that . Choose an integer such that .
    Then
    Consider the lowest common multiple of . This number will be of the form , where is an odd integer. Now multiply both sides of the equation by this number, to get

    Now, when multiplied out, all the terms on the left will be integers, except one:

    is not an integer, since is odd. So the left hand side is not an integer, and hence neither is the right hand side. That means that is not an integer.





    Not:
    http://plus.maths.org/issue12/features/harmonic/index.html
    Last edited by Suzan; October 12th 2008 at 11:11 AM.
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  15. #15
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    Quote Originally Posted by ThePerfectHacker View Post
    1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
    I'm confussed since \sum_{k=1}^{n}\frac{(-1)^{n+1}}{k}<1

    However H_n=\sum_{k=1}^{n}\frac{1}{k}\to\infty and my understanding is that H_n is never an integer for  n>1. This one would seem to be more interesting to prove.

    [edit] I think that's what Susan did. Never mind but perhaps we should make it explicit that's what's going on.
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