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How to find limit of a recursive sequence?

asked 2012-11-24 16:57:16 -0500

anonymous user

Anonymous

For example, I have this sequence: f(0) = 1 f(n) = 1/5 * (f(n-1)^2 + f(n-1) + 3) How do I find the limit of this sequence? I have been searching for how to do this for a while, but I can't find the answer.

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answered 2012-11-24 18:01:40 -0500

achrzesz gravatar image

updated 2012-11-24 21:12:45 -0500

In your example all f(n) are equal to 1, so the limit is 1 :)

In the case of linear recursive definitions you can use rsolve from sympy or solve_rec from Maxima, and then use limit function. In the general nonlinear case you can experiment numerically: http://ask.sagemath.org/question/1247....

Using Maxima you can do:

sage: maxima('f[n]:=1/5*(f[n-1]^2+f[n-1]+3)')
f[n]:=(3+f[n-1]+f[n-1]^2)/5
sage: maxima('f[0]:1')                       
1
sage: maxima('f[100]')                       
1
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Asked: 2012-11-24 16:57:16 -0500

Seen: 917 times

Last updated: Nov 24 '12