Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series.
In 2006, Jonathan Sondow gave a nice geometric proof that e is irrational. Moreover, he said that a generalization of his construction may be used to prove the Cantor's theorem. But, he did not do that in his paper, see [2]. So we give a geometric proof to Cantor's theorem using a generalization to Sondow's construction. After, it is given an irrationality measure for some Cantor series, for that we generalize the Smarandache function. Also we give an irrationality measure for e that is a bit better than the given one in [2].
is called Cantor series.
The number e is a Cantor series. For see that, take
We recall the following theorem due to Cantor [1].
Theorem (Cantor) Let θ be a Cantor series. Suppose that each prime divides infinitely many of the b n . Then θ is irrational if and only if both a n > 0 and a n < b n -1 hold infinitely often.
Proof For proving the necessary condition, observe that if a n = 0 for n ≥ n 0 , then the series is a finite sum, hence θ is rational. If a n > 0 infinitely often, let us to construct a nested sequence of closed intervals I n with intersection θ.
Proceeding inductively, we have two possibilities, the first one, if a n = 0, so define I n = I n-1 . When a n = 0, divide the interval
where A n ∈ Z for each n ∈ N. Also θ ∈ I n for all n ≥ 1. In fact, by hypothesis it is easy see that θ > An b 1 •••bn , for all n ≥ 1. For the other inequality, note that
Now if a n = b n -1 for n ≥ n 0 , then θ is the right-hand endpoint of I n 0 -1 , because each I n contains that endpoint and the lengths of the I n tend to zero. Hence again θ is rational. For showing the sufficient condition, note that if a m < b m -1, then holds the strict inequality in (2), for each n < m. Since a n > 0 holds infinitely often,
Suppose that θ = p q ∈ Q. Each prime number divides infinitely many b n , so there exist n 0 sufficiently large such that q|b
Hence θ lies in interior of I N . Also I N = I n 0 +k for some k ≥ 0. Suppose
. But that is again a contradiction. Therefore, it follows the irrationality of θ. ✷
The next step is to give an irrationality measure for some Cantor series. Now, we construct an uncountable family of functions, where one of them is exactly a well-known function for us. In [2], J. Sondow showed that for all integers p and q with q > 1,
where S(q) is the smallest positive integer such that S(q)! is a multiple of q (the so-called Smarandache function, see [3]). Note that if η = (1, 2, 3, …), then D(q, η) = S(q). Since e is a Cantor series and D(•, σ) is a generalization of Smarandache function, it is natural to think in a generalization or an improvement to the inequality in (3).
for all m ∈ Z.
Proof Suppose that the result fail for some m ∈ Z. So, m b 1 •••bn lies in interior of I n . Contradiction. Hence (4) holds for all m ∈ Z. ✷ Proposition Suppose that a Cantor series θ, like in (1) and satisfying ( * ), is an irrational number. For all integers p ∈ Z and q ∈ Z * , with D(q, σ) > 1, let k be the smallest integer greater than D(q, σ) such that the interval I k lies in the interior of I D(q,σ) . Then
where σ = (b 1 , b 2 , …).
Proof Let σ = (b 1 , b 2 , …). Set n = D(q, σ) and m = pb . Therefore m and n are integers and
The inequalities (6) and (7) follow respectively by Lemma 1 and the hypothesis on k.
The result below gives a slight improvement to (3).
Corollary If p and q are integers, with q = 0, then
where σ = (2, 3, 4, …).
Proof Since that min p∈Z |e -p| > 0.28 > 1 6 , then (8) holds in the case q = ±1. In case q = ±1 the inequality also holds by Proposition and Example 1. Moreover, in this case we have S(q)-1 ∈ {n ∈ N | q|(n+1)!} and D(q, σ)+1 ∈ {n ∈ N | q|n!}. Thus S(q) = D(q, σ) + 1. Hence e -p q > 1 (D(q, σ) + 2)! = 1 (S(q) + 1)! ✷ Actually, the improvement happens only because (8) also holds for q = ±1.
The number ξ := 1 (1!) 5 + 1 (2!) 5 + 1 (3!) 5 +. . . = 1.031378… is irrational, moreover for p, q ∈ Z with q = 0, we have ξ -p q > 1 (D(q, σ) + 2)! 5 where σ = (2 5 , 3 5 , …).
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