# Problem 48

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• Apr 10th 2008, 09:20 PM
ThePerfectHacker
Problem 48
1) Let $n\geq 2$ prove that $1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
• Apr 21st 2008, 01:09 PM
Math's-only-a-game
Quote:

Originally Posted by ThePerfectHacker
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. (Hi)
• Apr 23rd 2008, 02:14 PM
icemanfan
By the alternating series theorem, the partial sum will always be less than one but greater than zero, and therefore not an integer.
• Apr 25th 2008, 12:31 PM
Aryth
Quote:

Originally Posted by ThePerfectHacker
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.
• Apr 25th 2008, 12:36 PM
1 +1/2 + 1/3 + 1/4 + .... ===>A
and
1/2 + 1/4 + 1/6 +.... =====>B

to get the required series:
A - 2B
• Apr 25th 2008, 01:14 PM
Aryth
Yeah, that is a distinct possibility...

The Alternating Series does equal:

H(n) - H(2n)

Where H(n) is the n-th harmonic number
• May 14th 2008, 06:11 AM
Henderson
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?
• Jun 4th 2008, 10:01 AM
Aryth
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.
• Jun 6th 2008, 12:14 PM
Jacobsen
[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]
• Jul 7th 2008, 03:14 PM
meymathis
Quote:

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.
• Jul 7th 2008, 08:06 PM
meymathis

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:

$
O_N = 1 - \frac{1}{2} + \ldots + \frac{1}{2N+1}
$
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:

$
E_N = 1 + \ldots - \frac{1}{2N}
$
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.
• Jul 30th 2008, 05:58 AM
Lore
Quote:

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...
• Sep 8th 2008, 09:59 AM
bkarpuz
Quote:

Originally Posted by Lore
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.
• Oct 10th 2008, 10:49 PM
Suzan
Quote:

Originally Posted by ThePerfectHacker
1) Let $n\geq 2$ prove that $1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.

Suppose that http://plus.maths.org/MI/plus/issue1...s/img-0001.png. Choose an integer http://plus.maths.org/MI/plus/issue1...s/img-0002.png such that http://plus.maths.org/MI/plus/issue1...s/img-0003.png.
Then http://plus.maths.org/MI/plus/issue1...s/img-0004.png
Consider the lowest common multiple of http://plus.maths.org/MI/plus/issue1...s/img-0005.png. This number will be of the form http://plus.maths.org/MI/plus/issue1...s/img-0006.png, where http://plus.maths.org/MI/plus/issue1...s/img-0007.png is an odd integer. Now multiply both sides of the equation by this number, to get
http://plus.maths.org/MI/plus/issue1...s/img-0008.png
Now, when multiplied out, all the terms on the left will be integers, except one:
http://plus.maths.org/MI/plus/issue1...s/img-0009.png
is not an integer, since http://plus.maths.org/MI/plus/issue1...s/img-0007.png is odd. So the left hand side is not an integer, and hence neither is the right hand side. That means that http://plus.maths.org/MI/plus/issue1...s/img-0010.png is not an integer.

Not:
http://plus.maths.org/issue12/features/harmonic/index.html
• Oct 12th 2008, 07:48 AM
shawsend
Quote:

Originally Posted by ThePerfectHacker
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.

 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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