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Topic: logarithm reciprocal limit
Replies: 8   Last Post: Aug 11, 2010 7:30 AM

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 bo198214 Posts: 83 Registered: 6/11/06
logarithm reciprocal limit
Posted: Jul 20, 2010 3:00 AM

In my current research I encountered the following intriguing
sequence:

Define a_n recursively in the following way (for any b>0):

a_1 = 1 and for n>1:
a_n = 1/(1-b^n) * sum from m=1 to n-1 over a_m * (n over m) * (1-b)^(n-
m) * b^n

For ASCII handicapped people here the typeset formula:
http://math.eretrandre.org/cgi-bin/mimetex.cgi?a_n%20=%20\frac{1}{1-b^n}\sum_{m=1}^{n-1}%20a_m%20\left(n\\m\right)%20(1-b)^{n-m}%20b^m

My conjecture is that
a_n -> (b-1)/ln(b)

Have anyone heard about such a formula, or can prove or disprove its
convergence?
It sounds too elementary for not being treated somewhere already.

Date Subject Author
7/20/10 bo198214
7/20/10 William Elliot
7/20/10 bo198214
7/23/10 bo198214
7/24/10 Roland Franzius
7/24/10 bo198214
8/11/10 bo198214