A New Formula for the Natural Logarithm of a Natural Number
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A New Formula for the Natural Logarithm of a Natural Number Shahar Nevo
Received: 12 September 2012 / Revised: 29 November 2012 / Accepted: 29 January 2013 / Published online: 23 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract For every natural number T, we write Ln T as a series, generalizing the known series for Ln 2. We also introduce related linear subspaces of C. Keywords Euler-Mascheroni constant · Natural logarithm · Infinite series · Definite integral · Linear spaces Mathematics Subject Classification (2000)
26A09 · 40A05 · 40A30
1 Introduction Notation The letters Q, R and C denote, respectively, the fields of rational numbers, of real numbers and of complex numbers. The Euler–Mascheroni constant γ , [1], [2, p. 18], is given by the limit γ = lim An , n→∞
(1)
where for every n ≥ 1, An := 1 + 21 + · · · + n1 − Ln n. In Sect. 2, we prove Theorem 1, which presents every natural logarithm of a natural number by a series, that is, a rearrangement of the conditionally convergent series
Communicated by Stephan Ruscheweyh. This research is supprted by the Israel Science Foundation, Grant No. 395/07. S. Nevo (B) Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel e-mail: [email protected]
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S. Nevo
1 − 1 + 21 − 21 + 13 − 13 + · · · . This series generalizes the known series Ln 2 = 1 − 21 + 13 − 41 + · · · and is achieved using (1). The nature of the general term in this series motivated us to study certain linear spaces in C, F (T ), where F ⊂ C is a field and T ≥ 2 is an integer. The study of these linear spaces is the content of Sect. 3 where Theorem 2 supplies basic information about the dimension of F (T ). In Sect. 4, we introduce a different Proof of Theorem 1, using an elementary integral calculus. This proof is a little more complicated than the first one, but this proof makes it possible for us to calculate members of F (T ) by some definite integrals. In Sect. 5, we give a few basic examples for such calculations. 2 The New Formula Theorem 1 Let T ≥ 2 be an integer, then Ln T =
∞ k=0
1 1 (T − 1) 1 + + ··· + − . kT + 1 kT + 2 kT + (T − 1) kT + T
(2)
∞ Proof First let us prove ∞ the convergence of {An }n=1 from (1). Set A0 := 0, and consider the series n=0 (An+1 − An ). By Lagrange’s Mean Value Theorem, there exists for every n ≥ 1 a number θn , 0 < θn < 1 such that
An+1 − An =
1 1 1 θn − 1 − Ln(n + 1) + Ln n = − , = n+1 n + 1 n + θn (n + 1)(n + θn )
and thus 0 > An+1 − An > Euler–Mascheroni constant). We have AnT =
−1 n(n+1)
n−1 T k=0 j=1
and the series converges to some limit γ (the
1 − Ln(nT ) → γ . n→∞ kT + j
(3)
By subtracting (1) from (3) and using Ln(nT ) = Ln n + Ln T, we get ⎛ n−1 T ⎝ k=0
j=1
⎞ 1 ⎠ 1 − → Ln T, kT + j k + 1 n→∞
that is, Ln T =
∞ k=0
as required.
123
1 1 1 (T − 1) + + ··· + − , kT + 1 kT + 2 kT + (T − 1) kT + T
A New Formula for the Natural Logarithm
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By Theorem 1, we have, for example, 1 1 2 1 1 2 1 1 2 + + + ··· Ln 3 = + − + − + − 1 2 3 4 5 6 7 8 9
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