I am trying to figure out how can i get limit of recursive sequence.
For example this one.
Is there any common way of solving all recursive limits?
IF the limit exists, then take the limit of both sides of the recursion equation: . If the limit exists and , then both of those limits are a. You get the equation . Solve that equation for a.
Of course, that only works if the sequence converges. Here you can see that the first few numbers are 1, 3/2, 7/4, .... You should be able to prove, by induction, that this sequence is increasing and has 2 as an upper bound. By the "monotone sequence theorem", then, the sequence converges.
Recursive sequences, because they give an explicit formula for in terms of , are easier to use with induction!
For example, to prove this sequence is increasing:
1) If n= 1, and so .
2) Assume that, for some k, . Then .
(I have used the fact that, since , .)
Its just as easy to prove that 2 is an upper bound for the sequence.