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I What is this sequence that converges to ln(x) called?

  1. Nov 21, 2017 #1
    I found the following convergent sequence for the natural logarithm online: [tex]\lim_{a\rightarrow\infty}a x^{1/a}-a=\ln(x)[/tex] Does anybody know where this sequence first appeared, or if it has a name?
     
  2. jcsd
  3. Nov 21, 2017 #2

    fresh_42

    Staff: Mentor

    There is an attempt to list known integer sequences: http://oeis.org/ but I doubt that this one has a certain name. Alfred Hurwitz has found these two sequences with ##\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = \ln x##
    $$\begin{aligned}
    a_{n}&=2^{n}(x_{n}-1)\\b_{n}&=2^{n}(1-1/x_{n})\end{aligned}$$
    which thus could be called Hurwitz sequences. Wikipedia says (which means I haven't checked) ##\ln x = \lim_{n \to \infty} n \left( 1 - \frac{1}{\sqrt[n]{x}} \right) ## is equivalent to ## \ln x= \lim_{h \to 0} \frac{x^{h}-1}{h} = \lim_{h \to 0} \int_{1}^{x}\frac{1}{t^{1-h}}\, dt## by L’Hôpital's rule. So the sequence you mentioned is in a way the natural definition, given that ##\ln |x| = \int \frac{1}{x}\,dx##
     
  4. Nov 22, 2017 #3

    Erland

    User Avatar
    Science Advisor

    The limit statement is equivalent to
    $$\lim_{h\to 0} \frac{c^h-1}h=\ln c$$
    which gives the derivative of ##c^x## at ##x=0##.
     
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