Prove that infinite sequence defined recursively $a_{n+2}=\frac{1}{2}(a_{n+1}+a_{n})$ has limit and calculate it.
We know that $a_{1}=0 , a_{2}=1$.
I truly do not know how to proceed further. My teacher gave us a tip that we should separate it to even and odd partial series. But I wasn't able to get something useful from that.
Thanks for help:)
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John Doe
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3 Answers
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Yes, try to separate even and odd terms. You should use the theorem that states if you have an increasing/decreasing bounded sequence then the limit exists.
So, try to show that the odd and even sub-sequences of $\{ a_n \}_{n \in \mathbb{N}}$ form an/a increasing/decreasing sequence and see if you can show that they are bounded or not.
user66733
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To calculate the limit, you should first solve the recurrence relation and find $a_n$ I think. This is usually covered in a course about discrete mathematics or linear algebra, I'm not sure if it's taught in analysis or calculus. But the existence of the limit can be shown that way. – user66733 Nov 30 '13 at 11:32
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${ a_n }_{n \in \mathbb{N}}$ is neither increasing or decreasing, but the odd / even partial series definitely are. – chx Nov 30 '13 at 12:52
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@chx: Yes, I meant the same thing. I'll fix it. – user66733 Nov 30 '13 at 12:54
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I already put that answer in :) – chx Nov 30 '13 at 12:55
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If you want to solve it with Maple, see this answer from me. You will first get an explicit form of that recurrence relation and then you can deduce the limit from it.
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The partial series are monotonic (one increasing, one decreasing) and bound. They have limits. Consider whether these limits can be different.
chx
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