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

1. Nov 21, 2017

### Daniel Gallimore

I found the following convergent sequence for the natural logarithm online: $$\lim_{a\rightarrow\infty}a x^{1/a}-a=\ln(x)$$ Does anybody know where this sequence first appeared, or if it has a name?

2. Nov 21, 2017

### 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 $\ln |x| = \int \frac{1}{x}\,dx$

3. Nov 22, 2017

### Erland

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 $x=0$.