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Expression Ratio Limit (Posted on 2015-10-22) Difficulty: 3 of 5
Given that x is a positive real number, and:
A(x,n) = (x+1).....(x + n)


Limit Σn=1 to p (n-1)!/A(x,n)
p → ∞

No Solution Yet Submitted by K Sengupta    
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Stumped Comment 1 of 1
A(x,n) = (x+1).....(x + n), which I assume is the product of all integers x+1 to x+n.  Then A(n) = n!/(x+n)!

Then the limit is an infinite sum with each term equal to (n-1)!*n!/(x+n)!:
0!*1!/(x+1)! + 1!*2!/(x+2)! + 2!*3!/(x+3)! + 3!*4!/(x+4)! + ...

The ratio of term n-1 to term n is (n-1)*n/(x+n). For any value x there is a value for n when this fraction exceeds 1, any n larger than this value also make the fraction exceed 1.  So for any specific x, the sum is divergent.

I'm not sure what to make of this as the problem seems to imply there is a closed form.

  Posted by Brian Smith on 2017-06-24 12:11:00
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