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Computing a specific sum

  1. Mar 12, 2014 #1
    Suppose we have a Markov chain with stationary distributions ##p_n=\frac{a}{nb+c}p_{n-1}## for ##n\in\mathbb{N}## where ##a,b## and ##c## are some positive constants.
    It follows that ##p_n=p_0\prod_{i=1}^n\frac{a}{ib+c}##. Normalisation yields ##1=p_0\sum_{n=0}^\infty\prod_{i=1}^n\frac{a}{ib+c}## so ##p_0=\left(\sum_{n=0}^\infty\prod_{i=1}^n\frac{a}{ib+c}\right)^{-1}##.

    Question: how can one compute the sum in the brackets?
     
    Last edited: Mar 12, 2014
  2. jcsd
  3. Mar 27, 2014 #2
    Well, let's think about this. What can we do to simplify $$\prod_{1\leq i\leq n}\frac{a}{bi+c}$$ into "workable" terms? Can you give us an attempt?
     
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