1) Let prove that 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.
The series' terms look like such:
This series converges to
Since the series converges to and since:
Then for the series can never reach one since it is incrementing up or down by smaller amounts. Since you subtract 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 because all terms are decreasing,therefore the partial sums remain between 1 and 0.
[FONT='Cambria Math','serif']My first thought was to try an inductive argument, but I had a lot of difficulty getting it going. I dont 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
Let Pn denote the proposition thatn≥2; 1-1/2+1/3- ±1/n ∈ (0,1).
and1-1/2+1/3- ±1/(n-1) ∈ (0,1)
Then P3 is true since1-1/2+1/3- ±1/(n-1)±1/n∈ (0,1).
Assume Pn is true and that n is even. Then1-1/2+1/3=5/6∈ (0,1)
and1-1/2+1/3- +1/(n-1) ∈ (0,1)
Because 1/(n+1) < 1/n, it follows from the inductive hypothesis that1-1/2+1/3- +1/(n-1)-1/n ∈ (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. //1-1/2+1/3- +1/(n-1)-1/n+1/(n+1) ∈ (0,1).
Ok, how about this. It is basically Aryth's post but a bit more explicit.
Let which is just the partial sums.
Consider the (sub) sequence of partial sums:
is a subsequence of which as noted above converges to ln(2) (derive using MacLauren expansion of ln at x=1). Then .
is monotonically decreasing:
Note that and so for and so cannot be an integer.
Likewise for the partial sums:
except that monotonically increases from 1/2 to ln(2).
Put it together and we just showed the odd and even elements of the partial sums are never integers after 1.
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.