Let the sequence \( (a_n)_{n\in\mathbb{N}} \) be defined recursively in the following way for \( b>0 \):
\( a_1 = 1 \) and \( a_n = \frac{1}{1-b^n}\sum_{m=1}^{n-1} a_m \left(n\\m\right) (1-b)^{n-m} b^m \) for \( n\ge 2 \)
Is \( \lim_{n\to\infty} a_n = \frac{b-1}{\ln(b)} \)?
The conjecture is proven to be true.
See
http://arxiv.org/abs/1008.1409
and also the discussions:
Logarithm reciprocal
True or false logarithm
There is also a result-less (as of this writing) thread in sci.math.research and sci.math called "Logarithm reciprocal".
\( a_1 = 1 \) and \( a_n = \frac{1}{1-b^n}\sum_{m=1}^{n-1} a_m \left(n\\m\right) (1-b)^{n-m} b^m \) for \( n\ge 2 \)
Is \( \lim_{n\to\infty} a_n = \frac{b-1}{\ln(b)} \)?
The conjecture is proven to be true.
See
http://arxiv.org/abs/1008.1409
and also the discussions:
Logarithm reciprocal
True or false logarithm
There is also a result-less (as of this writing) thread in sci.math.research and sci.math called "Logarithm reciprocal".